ALLEN Tangent Normal PDF

ALLEN Tangent Normal 1...

ALLEN Tangent Normal 1 RATE MEASURE, TANGENT & NORMAL 1. RATE MEASUREMENT : Whenever one quantity y varies with another quantity x, satisfying some rule y =ƒ(x), then dy dy ù (or ƒ'(x)) represents the rate of change of y with respect to x and (or ƒ '(a)) represents the dx dx úû x =a rate of change of y with respect to x at x = a. Illustration 1 : The volume of a cube is increasing at a rate of 9cm3/s. How fast is the surface area increasing when the length of an edge is 10cm ? Solution : Let x be the length of side, V be the volume and S be the surface area of the cube. Then V= x3 and S = 6x2, where x is a function of time t. dV d dx = 9cm 3 / s = (x3 ) = 3x 2 dt dt dt dx 3 Þ = dt x 2 dS d æ 3 ö 36 = (6x 2 ) = 12x ç 2 ÷ = dt dt èx ø x dS ù 2 dt úû x =10cm = 3.6 cm /s. Illustration 2 : x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of the second square with respect to the first square. Solution : Given x and y are sides of two squares. Thus the area of two squares are x2 and y2 dy d ( y2 ) dx = y. dy 2y We have to obtain =........ (i) d (x ) node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 2 2x x dx dy where the given curve is, y = x – x2 Þ = 1 - 2x........ (ii) dx d ( y2 ) y( Thus, = 1 - 2x ) [from (i) and (ii)] d ( x2 ) x d (y2 ) (x - x 2 )(1 - 2x) d (y2 ) or = Þ = (2x 2 - 3x + 1) d (x2 ) x d (x2 ) The rate of change of the area of second square with respect to first square is (2x2 – 3x + 1) E 2 JEE-Mathematics ALLEN Do yourself - 1 : (i) What is the rate of change of the area of a circle with respect to its radius r at r = 6cm. (ii) A stone is dropped into a quiet lake and waves move in circles at the speed of 5cm/s. At the instant when the radius of the circular wave is 8cm, how fast is the enclosed area increasing ? 2. APPROXIMATION USING DIFFERENTIALS : In order to calculate the approximate value of a function, differentials may be used where the differential of a function is equal to its derivative multiplied by the differential of the independent variable. In general dy = f '(x)dx or df(x) = f '(x)dx Note : (i) For the independent variable ‘x’, increment D x and differential dx are equal but this is not the case with the dependent variable ‘y’ i.e. Dy ¹ dy. \ Approximate value of y when increment Dx is given to independent variable x in y = f(x) is dy y + Dy = f(x + Dx) = f(x) +.Dx dx dy (ii) The relation dy = f '(x) dx can be written as = f '(x) ; thus the quotient of the differentials of dx ‘y’ and ‘x’ is equal to the derivative of ‘y’ w.r.t. ‘x’. Illustration 3 : Find the approximate value of square root of 25.2. Solution : Let f(x) = x Dx Now, f(x + Dx) – f(x) = f '(x). Dx = 2 x we may write, 25.2 = 25 + 0.2 Taking x = 25 and Dx = 0.2, we have node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 0.2 f(25.2) – f(25) = 2 25 0.2 or f(25.2) – 25 = = 0.02 Þ f(25.2) = 5.02 10 or ( 25.2 ) = 5.02 Do yourself - 2 : (i) Find the approximate value of (0.009)1/3. E  ALLEN Tangent Normal 3 3. TANGENT TO THE CURVE AT A POINT : The tangent to the curve at 'P' is the line through P whose slope is limit of the secant slopes as Q ® P from either side. 4. MYTHS ABOUT TANGENT : (a) Myth : A line meeting the curve only at one point is a tangent to the curve. Explanation : A line meeting the curve in one point is not necessarily tangent to it. Here L is not tangent to C (b) Myth : A line meeting the curve at more than one point is not a tangent to the curve. Explanation : A line may meet the curve at several points and may still be tangent to it at some point Here L is tangent to C at P, and cutting it again at Q. (c) Myth : Tangent at a point to the curve can not cross it at the same point. node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 Explanation : A line may be tangent to the curve and also cross it. Here X-axis is tangent to y = x3 at origin. 5. NORMAL TO THE CURVE AT A POINT : A line which is perpendicular to the tangent at the point of contact is called normal to the curve at that point. E 4 JEE-Mathematics ALLEN 6. POINTS TO REMEMBER : (a) The value of the derivative at P(x1, y1) gives the slope of the tangent to the curve at P. Symbolically dy ù f ' ( x1 ) = = Slope of tangent at P(x1, y1) = m(say). dx úû (x1 , y1 ) dy ù (b) Equation of tangent at (x1, y1) is ; y - y1 = (x - x1 ) dx úû (x1 , y1 ) 1 (c) Equation of normal at (x1, y1) is ; y – y1 = - (x - x1 ). dy ù dx úû (x1 ,y1 ) Note : (i) The point P (x1, y1) will satisfy the equation of the curve & the equation of tangent & normal line. (ii) If the tangent at any point P on the curve is parallel to the axis of x then dy/dx = 0 at the point P. (iii) If the tangent at any point on the curve is parallel to the axis of y, then dy/dx not defined or dx/dy = 0. (iv) If the tangent at any point on the curve is equally inclined to both the axes then dy/dx = ±1. (v) If a curve passing through the origin be given by a rational integral algebraic equation, then the equation of the tangent (or tangents) at the origin is obtained by equating to zero the terms of the lowest degree in the equation. e.g. If the equation of a curve be x2 – y2 + x3 + 3x2y –y3=0, the tangents at the origin are given by x 2 – y2 = 0 i.e. x + y = 0 and x – y = 0 Illustration 4 : Find the equation of the tangent to the curve y = ( x 3 - 1) ( x - 2 ) at the points where the curve cuts the x-axis. Solution : The equation of the curve is y = ( x 3 - 1) ( x - 2 ).......... (i) node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 It cuts x-axis at y = 0. So, putting y = 0 in ( i ) , we get ( x3 - 1) ( x - 2 ) = 0 ( ) Þ ( x - 1)( x - 2 ) x 2 + x + 1 = 0 Þ x - 1 = 0, x - 2 = 0 éëQ x 2 + x + 1 ¹ 0 ùû Þ x = 1, 2. Thus, the points of intersection of curve (i) with x-axis are (1, 0) and (2, 0). Now, dy æ dy ö æ dy ö ( ) y = x3 - 1 ( x - 2 ) Þ dx ( ) = 3x 2 ( x - 2 ) + x3 - 1 Þ ç ÷ = -3 and ç ÷ è dx ø( 2,0 ) =7 è dx ø(1,0 ) The equations of the tangents at (1, 0) and (2, 0) are respectively y - 0 = -3 ( x - 1) and y - 0 = 7 ( x - 2 ) Þ y + 3x - 3 = 0 and 7x - y - 14 = 0 Ans. E  ALLEN Tangent Normal 5 p Illustration 5 : The equation of the normal to the curve y = x + sin x cos x at x = is - 2 (A) x = 2 (B) x = p (C) x + p = 0 (D) 2x = p p p p æp pö Solution : Q x= Þ y = + 0 = , so the given point = ç , ÷ 2 2 2 è2 2ø dy æ dy ö Now from the given equation = 1 + cos2 x - sin 2 x Þ ç ÷ = 1 + 0 -1 = 0 dx è dx øæç p , p ö÷ è2 2ø p p Þ The curve has vertical normal at æç , ö÷. è2 2ø p The equation to this normal is x = 2 p Þ x - = 0 Þ 2x = p Ans. (D) 2 Illustration 6 : The equation of normal to the curve x + y = x y , where it cuts x-axis is - (A) y = x + 1 (B) y = - x + 1 (C) y = x - 1 (D) y = - x - 1 Solution : Given curve is x + y = x y..... (i) at x-axis y=0, \ x + 0 = x0 Þ x=1 \ Point is A(1, 0) Now to differentiate x + y = x y take log on both sides 1 ì dy ü 1 dy Þ log ( x + y ) = y log x \ í1 + ý = y. + ( log x ) x + y î dx þ x dx ì dy ü æ dy ö Putting x = 1, y = 0 í1 + ý = 0 Þ ç ÷ = -1 î dx þ è dx ø(1,0 ) \ slope of normal = 1 y-0 Equation of normal is, =1 Þ y = x -1 Ans. (C) x -1 Do yourself - 3 : node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 (i) Find the distance between the point (1,1) and the tangent to the curve y = e2x + x2 drawn at the point where the curve cuts y-axis. (ii) Find the equation of a line passing through (–2,3) and parallel to tangent at origin for the circle x2 + y2 + x – y = 0. 7. ANGLE OF INTERSECTION BETWEEN TWO CURVES : y Angle of intersection between two curves is defined as the angle between the two tangents drawn to the two curves at their point q of intersection. x O E 6 JEE-Mathematics ALLEN Orthogonal curves : If the angle between two curves at each point of intersection is 90° then they are called orthogonal curves. For example, the curves x2 + y2 = r2 & y = mx are orthogonal curves. Illustration 7 : The angle of intersection between the curve x 2 = 32y and y 2 = 4x at point (16, 8) is- æ3ö -1 æ 4 ö (A) 60° (B) 90° (C) tan -1 ç ÷ (D) tan ç ÷ è5ø è3ø dy x dy 2 Solution : x 2 = 32y Þ = Þ y 2 = 4x Þ = dx 16 dx y æ dy ö æ dy ö 1 \ at (16, 8) , ç ÷ = 1, ç ÷ = è dx ø1 è dx ø2 4 æ 1 ö ç 1- ÷ So required angle = tan -1 ç 4 ÷ = tan -1 æ 3 ö Ans. (C) ç5÷ ç 1 + 1æ 1 ö ÷ è ø ç ç4÷÷ è è øø Illustration 8 : Check the orthogonality of the curves y2 = x & x2 = y. Solution : Solving the curves simultaneously we get points of intersection as (1, 1) and (0, 0). æ dy ö 1 At (1,1) for first curve 2y ç ÷ = 1 Þ m 1 = y è dx ø1 2 (1,1) æ dy ö x & for second curve 2x = ç ÷ Þ m 2 = 2 O è dx ø 2 m1m2 ¹ –1 at (1,1). But at (0, 0) clearly x-axis & y-axis are their respective tangents hence they are orthogonal at (0,0) but not at (1,1). Hence these curves are not said to be orthogonal. Illustration 9 : If curve y = 1 - ax 2 and y = x 2 intersect orthogonally then the value of a is - 1 1 (A) (B) (C) 2 (D) 3 2 3 node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 dy dy Solution : y = 1 - ax 2 Þ = -2ax y = x2 Þ = 2x dx dx æ dy ö æ dy ö Two curves intersect orthogonally if ç ÷ ç ÷ = -1 è dx ø1 è dx ø2 Þ ( -2ax )( 2x ) = -1 Þ 4ax 2 = 1..... (i) Now eliminating y from the given equations we have 1 - ax 2 = x 2 Þ (1 + a ) x 2 = 1..... (ii) 4a 1 Eliminating x2 from (i) and (ii) we get =1 Þ a = Ans. (B) 1+ a 3 E  ALLEN Tangent Normal 7 Do yourself -4 : (i) If two curves y = ax and y = bx intersect at an angle a, then find the value of tana. (ii) Find the angle of intersection of curves y = 4 – x2 and y = x2. 8. LENGTH OF TANGENT, SUBTANGENT, NORMAL & SUBNORMAL : y = f(x) ent an g t h of t g Le n l en r ma no gth l P(x1, y1) of y y S a x T O M N dy Length tan Y = Length of subtangent of dx subnormal y1 1 + [f '(x1 )]2 (a) Length of the tangent (PT) = f '(x1 ) y1 (b) Length of Subtangent (MT) = f ' ( x1 ) Length of Normal (PN) = y1 1 + éë f ' ( x1 ) ùû 2 (c) (d) Length of Subnormal (MN) = |y1 f '(x1)| (e) Initial ordinate : Y intercept of tangent at point P(x, y) = OS (f) Radius vector (Polar radius) : Line segment joining origin to point P(x, y) = OP (g) Vectorial angle : Angle made by radius vector with positive direction of x-axis in anticlock wise direction is called vectorial angle. In given figure a is vectorial angle. p Illustration 10 : The length of the normal to the curve x = a ( q + sin q ) , y = a (1 - cos q ) at q = is - node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 2 a a (A) 2a (B) (C) 2a (D) 2 2 æ dy ö Solution : dy çè dq ÷ø a sin q q æ dy ö æpö = = = tan Þ ç ÷ = tan ç ÷ = 1 dx æ dx ö a (1 + cos q ) 2 è dx øq= p è4ø ç q÷ 2 èd ø p æ pö Also at q = , y = a ç 1 - cos ÷ = a 2 è 2ø 2 æ dy ö \ required length of normal = y 1 + ç ÷ = a 1 + 1 = 2a Ans. (C) è dx ø E 8 JEE-Mathematics ALLEN æ tö Illustration 11 : The length of the tangent to the curve x = a ç cos t + log tan ÷ , y = a sin t is è 2ø (A) ax (B) ay (C) a (D) xy dy æ dy ö æ dx ö a cos t Solution : =ç ÷ ç ÷ = = tan t dx è dt ø è dt ø æ 1 ö a ç - sin t + ÷ è sin t ø 2 æ dy ö 1+ ç ÷ è dx ø 1 + tan 2 t æ sec t ö \ length of the tangent = y = a sin t = a sin t ç ÷ = a Ans. (C) æ dy ö tan t è tan t ø ç dx ÷ è ø Do yourself - 5 : (i) Prove that at any point of a curve, the product of the length of sub tangent and the length of sub normal is equal to square of theordinates of point of contact. (ii) Find the length of subtangent to the curve x2 + y2 + xy = 7 at the point (1, –3). Miscellaneous Illustrations : Illustration 12 : Find the slope of normal at the point with abcissa x = –2 of the graph of the function ƒ(x) = |x2 – |x|| Solution : At x = –2, ƒ(x) becomes ƒ(x) = x2 + x dy = 2x + 1 = -3 dx 1 Slope of normal = 3 Illustration 13 : If y = 4x – 5 is a tangent to the curve y2 = px3 + q at (2, 3), then (A) p = 2, q = –7 (B) p = –2, q = 7 (C) p = –2, q = –7 (D) p = 2, q = 7 Solution : dy = 4 & 9 = 8p + q dx dy 2y = 3px 2 dx node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 dy dy 6 = 3p(4) Þ = 2p = 4 Þ p = 2 & q = –7 dx dx ANSWERS FOR DO YOURSELF 1: (i) 12p cm (ii) 80 pcm2/s 2: (i) 0.208 2 3: (i) units (ii) x – y + 5 = 0 5 lna - lnb æ4 2ö 4: (i) (ii) tan -1 çç ÷÷ 1 + lna lnb è 7 ø 5: (ii) 15 E  ALLEN Tangent Normal 9 TANGENT & NORMAL EXERCISE (O-1) 1. If the surface area of a sphere of radius r is increasing uniformly at the rate 8cm2/s, then the rate of change of its volume is : (A) proportional to r2 (B) constant (C) proportional to r (D) proportional to r TN0001 2. A Spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in the surface area (in cm2/min.) of the balloon when its diameter is 14 cm, is : (A) 10 (B) 10 10 (C) 100 (D) 10 TN0002 3. Let S be a square with sides of length x. If we approximate the change in size of the area of S by dA h· , when the sides are changed from x0 to x0 + h, then the absolute value of the error in our dx x =x0 approximation, is (A) h2 (B) 2hx0 (C) x 02 (D) h TN0003 4. The slope of the curve y = sinx + cos2x is zero at the point, where- p p (A) x = (B) x = (C) x = p (D) No where 4 2 TN0004 5. 2 3 2 The equation of tangent at the point (at , at ) on the curve ay = x is- 3 (A) 3tx – 2y = at3 (B) tx – 3y = at3 node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 (C) 3tx + 2y = at3 (D) None of these TN0005 6. The equation of normal to the curve y = x3 – 2x2 + 4 at the point x = 2 is- (A) x + 4y = 0 (B) 4x – y = 0 (C) x + 4y = 18 (D) 4x – y = 18 TN0006 7. The slope of the normal to the curve x = a(q – sinq), y = a(1 – cosq) at point q = p/2 is- (A) 0 (B) 1 (C) –1 (D) 1/ 2 TN0007 E 10 JEE-Mathematics ALLEN 8. The coordinates of the points on the curve x = a(q + sinq), y= a(1 – cosq), where tangent is inclined an angle p/4 to the x-axis are- æ æp ö ö æ æp ö ö æ æ p öö (A) (a, a) (B) ç a ç -1÷ , a ÷ (C) ç a ç + 1÷ , a ÷ (D) ç a, a ç + 1÷ ÷ è è2 ø ø è è2 ø ø è è 2 øø TN0008 9. Consider the curve represented parametrically by the equation x = t3 – 4t2 – 3t and y = 2t2 + 3t – 5 where t Î R. If H denotes the number of point on the curve where the tangent is horizontal and V the number of point where the tangent is vertical then (A) H = 2 and V = 1 (B) H = 1 and V = 2 (C) H = 2 and V = 2 (D) H = 1 and V = 1 TN0009 10. The line x/a + y/b = 1 touches the curve y = be–x/a at the point- (A) (0, a) (B) (0, 0) (C) (0, b) (D) (b, 0) TN0010 11. If the tangent to the curve 2y3 = ax2 + x3 at a point (a, a) cuts off intercepts p and q on the coordinates axes, where p2 + q2 = 61, then a equals- (A) 30 (B) –30 (C) 0 (D) ±30 TN0011 12. The sum of the intercepts made by a tangent to the curve x + y = 4 at point (4, 4) on coordinate axes is- (A) 4 2 (B) 6 3 (C) 8 2 (D) 256 TN0012 node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 x 2 y2 13. The curve x2 – y2 = 5 and + = 1 cut each other at any common point at an angle- 18 8 (A) p/4 (B) p/3 (C) p/2 (D) None of these TN0013 14. The lines tangent to the curve y3 – x2y + 5y – 2x = 0 and x4 – x3y2 + 5x + 2y = 0 at the origin intersect at an angle q equal to- p p p p (A) (B) (C) (D) 6 4 3 2 TN0014 E  ALLEN Tangent Normal 11 15. If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3 – 1, t Î R, meets the curve again at a point Q, then the coordinates of Q are : (A) (t2 + 3, t3 – 1) (B) (t2 + 3, –t3 – 1) (C) (16t2 + 3, – 64t3 – 1) (D) (4t2 + 3, –8t3 – 1) TN0015 16. The angle of intersection between the curves y2 = 8x and x2 = 4y – 12 at (2, 4) is- (A) 90° (B) 60° (C) 45° (D) 0° TN0016 17. The length of subtangent to the curve x2 + xy + y2 =7 at the point (1, –3) is- (A) 3 (B) 5 (C) 15 (D) 3/5 TN0017 (length of normal)2 18. For a curve is equal to - (length of tangent)2 (A) (subnormal)/(subtangent) (B) (subtangent)/(subnormal) (C) (subtangent × subnormal) (D) constant TN0018 19. At any point of a curve (subtangent) × (subnormal) is equal to the square of the- (A) slope of the tangent at that point (B) slope of the normal at that point (C) abscissa of that point (D) ordinate of that point TN0019 EXERCISE (O-2) [SINGLE CORRECT CHOICE TYPE] node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 1. Let S be a sphere with radius r. If we approximate the change of volume of S by h 2 dA h.A r + where A is surface area, when radius is changed from r0 to (r0 + h), then the 0 2 dr r = r0 absolute value of error in our approximation is 4p 3 (A) h3 (B) 4phr02 (C) 4pr0h2 (D) h 3 TN0020 E 12 JEE-Mathematics ALLEN 2. At the point P(a, an) on the graph of y = xn (n Î N) in the first quadrant a normal is drawn. The 1 normal intersects the y-axis at the point (0, b). If Lim b = , then n equals a ®0 2 (A) 1 (B) 3 (C) 2 (D) 4 TN0021 x2 3. A circle with centre at (15, –3) is tangent to y = at a point in the first quadrant. The radius of the 3 circle is equal to (A) 5 6 (B) 8 3 (C) 9 2 (D) 6 5 TN0022 x2 4. A line L is perpendicular to the curve y = - 2 at its point P and passes through (10, –1). The 4 coordinates of the point P are (A) (2, –1) (B) (6, 7) (C) (0, –2) (D) (4, 2) TN0023 5. The point(s) at each of which the tangents to the curve y = x3 - 3x2 - 7x + 6 cut off on the positive semi axis OX a line segment half that on the negative semi axis OY then the co-ordinates of the point(s) is/are given by : (A) (- 1, 9) (B) (3, - 15) (C) (1, - 3) (D) none TN0024 a b 6. The x-intercept of the tangent at any arbitrary point of the curve + = 1 is proportional to: x2 y2 (A) square of the abscissa of the point of tangency (B) square root of the abscissa of the point of tangency (C) cube of the abscissa of the point of tangency (D) cube root of the abscissa of the point of tangency. node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 TN0025 7. A curve is represented by the equations, x = sec2 t and y = cot t where t is a parameter. If the tangent at the point P on the curve where t = p/4 meets the curve again at the point Q then ½PQ½ is equal to: 5 3 5 5 2 5 3 5 (A) (B) (C) (D) 2 2 3 2 TN0026 8. A curve is represented parametrically by the equations x = t + eat and y = – t + when t Î R and eat a > 0. If the curve touches the axis of x at the point A, then the coordinates of the point A are (A) (1, 0) (B) (1/e, 0) (C) (e, 0) (D) (2e, 0) TN0027 E  ALLEN Tangent Normal 13 9. At any two points of the curve represented parametrically by x = a (2 cos t - cos 2t) ; y = a (2 sin t - sin 2t) the tangents are parallel to the axis of x corresponding to the values of the parameter t differing from each other by : (A) 2p/3 (B) 3p/4 (C) p/2 (D) p/3 TN0028 [MULTIPLE CORRECT CHOICE TYPE] x y K 10. If + = 1 is a tangent to the curve x = Kt, y = , K > 0 then : a b t (A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0 TN0029 x3 5 x2 11. The co-ordinates of the point(s) on the graph of the function, f(x) = - + 7x – 4 where the 3 2 tangent drawn cut off intercepts from the co-ordinate axes which are equal in magnitude but opposite in sign, is (A) (2, 8/3) (B) (3, 7/2) (C) (1, 5/6) (D) none TN0030 12. Given that g(x) is a non constant linear function defined on R- –1 –1 (A) y = g(x) and y = g (x) are orthogonal (B) y = g(x) and y = g (–x) are orthogonal (C) y = g(–x) and y = g–1(x) are orthogonal (D) y = g(–x) and y = g–1(–x) are orthogonal TN0031 2 2x 13. If the curves y = 2(x – a) and y = e touches each other, then 'a' is less than- (A) –1 (B) 0 (C) 1 (D) 2 TN0032 2x 14. For the curve C : y = e cosx, which of the following statement(s) is/are true ? (A) equation of the tangent where C crosses y-axis is y = 3x + 1 (B) equation of the tangent where C crosses y-axis is y = 2x + 1 æ p 3p ö (C) number of points in ç - , ÷ where tangent on the curve C is parallel to x-axis is 4. è 2 2 ø æ p 3p ö node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 (D) number of points in ç - , ÷ where tangent on the curve C is parallel to x-axis is 2. è 2 2 ø TN0033 EXERCISE (S-1) 1. Water is being poured on to a cylindrical vessel at the rate of 1 m3/min. If the vessel has a circular base of radius 3 m, find the rate at which the level of water is rising in the vessel. TN0034 2. A man 1.5 m tall walks away from a lamp post 4.5 m high at the rate of 4 km/hr. (i) how fast is the farther end of the shadow moving on the pavement ? (ii) how fast is his shadow lengthening ? TN0035 E 14 JEE-Mathematics ALLEN 3. A particle moves along the curve 6 y = x3 + 2. Find the points on the curve at which the y coordinate is changing 8 times as fast as the x coordinate. TN0036 4. An inverted cone has a depth of 10 cm & a base of radius 5 cm. Water is poured into it at the rate of 1.5 cm3/min. Find the rate at which level of water in the cone is rising, when the depth of water is 4 cm. TN0037 5. Water is dripping out from a conical funnel of semi vertical angle p/4, at the uniform rate of 2 cm3/sec through a tiny hole at the vertex at the bottom. When the slant height of the water is 4 cm, find the rate of decrease of the slant height of the water. TN0038 6. An air force plane is ascending vertically at the rate of 100 km/h. If the radius of the earth is R Km, how fast the area of the earth, visible from the plane increasing at 3min after it started ascending. 2 Take visible area A = 2pR h Where h is the height of the plane in kms above the earth. R +h TN0039 7. If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining elements are changed slightly, show that da + db = 0. cos A cos B TN0040 8. A water tank has the shape of a right circular cone with its vertex down. Its altitude is 10 cm and the radius of the base is 15 cm. Water leaks out of the bottom at a constant rate of 1cu. cm/sec. Water is poured into the tank at a constant rate of C cu. cm/sec. Compute C so that the water level will be rising at the rate of 4 cm/sec at the instant when the water is 2 cm deep. TN0041 9. Sand is pouring from a pipe at the rate of 12 cc/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always 1/6th of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm. TN0042 10. 2 A circular ink blot grows at the rate of 2 cm per second. Find the rate at which the radius is node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 6 22 increasing after 2 seconds. Use p =. 11 7 TN0043 11. Water is flowing out at the rate of 6 m3/min from a reservoir shaped like a hemispherical bowl of p 2 radius R = 13 m. The volume of water in the hemispherical bowl is given by V = · y (3R - y) 3 when the water is y meter deep. Find (a) At what rate is the water level changing when the water is 8 m deep. (b) At what rate is the radius of the water surface changing when the water is 8 m deep. TN0044 E  ALLEN Tangent Normal 15 3 12. (i) Use differentials to a approximate the values of ; (a) 36.6 and (b) 26. TN0045 (ii) If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its volume. TN0046 13. Find the equation of the normal to the curve y = (1 + x)y + sin-1 (sin2x) at x = 0. TN0047 14. Find all the lines that pass through the point (1, 1) and are tangent to the curve represented parametrically as x = 2t – t2 and y = t + t2. TN0048 7 15. The tangent to y = ax2 + bx + at (1, 2) is parallel to the normal at the point (–2, 2) on the curve 2 y = x2 + 6x + 10. Find the value of a and b. TN0049 41x 3 16. A line is tangent to the curve f (x) = at the point P in the first quadrant, and has a slope of 3 2009. This line intersects the y-axis at (0, b). Find the value of 'b'. TN0050 17. Find all the tangents to the curve y = cos (x + y), - 2p £ x £ 2p, that are parallel to the line x + 2y = 0. TN0051 18. The curve y = ax3 + bx2 + cx + 5 , touches the x - axis at P (- 2 , 0) & cuts the y-axis at a point Q where its gradient is 3. Find a , b , c. TN0052 19. Find the gradient of the line passing through the point (2,8) and touching the curve y = x3. TN0053 20. (a) Find the value of n so that the subnormal at any point on the curve xy = a n n + 1 may be node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 constant. TN0054 (b) Show that in the curve y = a. ln (x2 - a2), sum of the length of tangent & subtangent varies as the product of the coordinates of the point of contact. TN0055 x2 y2 x2 y2 21. (a) Show that the curves + = 1 & + = 1 (K1 ¹ K2) intersect a 2 + K1 b 2 + K1 a 2 + K2 b2 + K2 orthogonally. TN0056 (b) If the two curves C1 : x = y2 and C2 : xy = k cut at right angles find the value of k. TN0057 E 16 JEE-Mathematics ALLEN y 22. Show that the angle between the tangent at any point 'A' of the curve ln (x2 + y2) = C tan–1 and x the line joining A to the origin is independent of the position of A on the curve. TN0058 EXERCISE (S-2) RATE MEASURE AND APPROXIMATIONS 1. A variable D ABC in the xy plane has its orthocentre at vertex 'B' , a fixed vertex 'A' at the origin and 7x 2 the third vertex 'C' restricted to lie on the parabola y = 1 +. The point B starts at the point 36 (0, 1) at time t = 0 and moves upward along the y axis at a constant velocity of 2 cm/sec. How fast 7 is the area of the triangle increasing when t = sec. 2 TN0059 2. Find the equations of the tangents drawn to the curve y2 – 2x3 – 4y + 8 = 0 from the point (1, 2). TN0060 3. Find the point of intersection of the tangents drawn to the curve x2y = 1 – y at the points where it is intersected by the curve xy = 1 – y. TN0061 4. A function is defined parametrically by the equations 1 1 é 2t + t 2 sin if t ¹ 0 é sin t 2 if t ¹ 0 f(t) = x = ê t ] and g(t) = y = ê t ë 0 if t = 0 ë o if t = 0 Find the equation of the tangent and normal at the point for t = 0 if exist. TN0062 5. There is a point (p,q) on the graph of f(x) = x2 and a point (r,s) on the graph of g(x) = –8/x where p > 0 and r > 0. If the line through (p,q) and (r,s) is also tangent to both the curves at these points respectively then find the value of (p + q). node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 TN0063 6. Tangent at a point P1 [other than (0 , 0)] on the curve y = x3 meets the curve again at P2. The tangent at P2 meets the curve at P3 & so on. Show that the abscissae of P1, P2, P3,......... Pn, form a GP. Also area of D(P1 P2 P3 ) find the ratio. area of D(P2 P3 P4 ) TN0064 1 7. The chord of the parabola y = - a2x2 + 5ax - 4 touches the curve y = at the point x = 2 and is 1- x bisected by that point. Find 'a'. TN0065 E  ALLEN Tangent Normal 17 8. Show that the condition that the curves x2/3 + y2/3 = c2/3 & (x2/a2) + (y2/b2) = 1 may touch if c = a + b. TN0066 9. Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos2t ; y = - a cos3t contained between the co-ordinate axes is equal to 2a. TN0067 EXERCISE (JM) 4 1. The equation of the tangent to the curve y = x + , that is parallel to the x-axis, is :- x2 [AIEEE-2010] (1) y = 0 (2) y = 1 (3) y = 2 (4) y = 3 TN0068 2. The normal to the curve, x2 + 2xy – 3y2 = 0, at (1, 1) : [JEE-MAIN 2015] (1) meets the curve again in the third quadrant (2) meets the curve again in the fourth quadrant (3) does not meet the curve again (4) meets the curve again in the second quadrant TN0069 æ 1 + sin x ö p 3. Consider f(x) = tan–1 çç ÷÷ , x Î æç 0, p ö÷. A normal to y = f(x) at x = also passes through è 1 - sin x ø è 2ø 6 the point : [JEE-MAIN 2016] æp ö æ 2p ö æp ö (1) ç ,0 ÷ (2) (0, 0) (3) ç 0, ÷ (4) ç ,0 ÷ è4 ø è 3 ø è6 ø TN0070 4. The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point : [JEE-MAIN 2017] æ1 1ö æ 1 1ö æ1 1ö æ1 1ö (1) ç , ÷ (2) ç - , - ÷ (3) ç , ÷ (4) ç , - ÷ node06\B0B0-BA\Kota\JEE(Advanced)\Module Coding (V-Tag)\Enthuse\Maths\AOD\Eng\01_Tangent Normal.p65 è 2 3ø è 2 2ø è2 2ø è2 3ø TN0071 5. If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is : 7 9 (1) (2) 4 (3) (4) 6 [JEE-MAIN 2018] 2 2 TN0072 2 6. The tangent to the curve, y = xe x passing through the point (1,e) also passes through the point : [JEE-MAIN 2019] æ4 ö æ5 ö (1) ç ,2e ÷ (2) (2,3e) (3) ç ,2e ÷ (4) (3,6e)