22 Questions
If the curve y = 1 – ax2 and y = x2 intersect orthogonally, then a is equal to
1/12
The normal of the curve x = a(cos θ + θ sin θ) and y = a(sin θ - θ cos θ) at any θ is such that
It is at a constant distance from the origin
The area of a triangle formed by a tangent to the curve 2xy = a2 and the coordinate axes is
3a2
If the curves y = a x and y = ex intersect at an angle α, then tan α equals
loge a + 1
What is the condition for the curves y = 1 – ax2 and y = x2 to intersect orthogonally?
a = 1/12
What is the normal vector of the curve x = a(cos θ + θ sin θ) and y = a(sin θ - θ cos θ) at any θ?
It is at a constant distance from the origin
What is the area of a triangle formed by a tangent to the curve 2xy = a2 and the coordinate axes?
3a2
If the curves y = a x and y = ex intersect at an angle α, what is the value of tan α?
loge a + 1
What is the equation of the tangent to the curve y = e^–|x| at the point where the curve cuts the line x = 1?
e^(x + y) = 1
What is the distance between the origin and the normal to the curve y = e^(2x) + x^2 at x = 0?
3/2
What is the value of (9m^2 + 2) if the tangent at any point P(4m^2, 8m^3) of the curve x^3 – y^2 = 0 is a normal also to the curve x^3 – y^2 = 0?
1
What is the value of c such that the line joining the points (0, 3) and (5, –2) becomes tangent to the curve y = c/(x + 1)?
2
What is the value of k if the gradient at B is k times the gradient at A, where the tangent at A except (0, 0) meets the curve again at B?
9
What is the equation of the curve where the tangent at A except (0, 0) meets the curve again at B?
y = x^3
How many tangents can be drawn from the point (2, 0) to the curve y = x^6?
1
What is the equation of the curve in parametric form, given x = sec^2 t and y = cot t?
x = sec^2 t, y = cot t
How many tangents to the curve y^2 – 2x^3 – 4y + 8 = 0 pass through the point (1, 2)?
1
If the curves x^2 + y^2 = 1 and y^3 = 16x intersect at right angles, what is the value of 3a^2?
1
What is the value of |PQ|, where P is the point on the curve x = sec^2 t and y = cot t at t = π/4, and Q is the point where the tangent at P meets the curve again?
2
How many points of intersection are there between the curves x^2 + y^2 = 1 and y^3 = 16x?
3
What is the equation of the tangent to the curve y = x^6 at the point (2, 0)?
y = 0
What is the slope of the tangent to the curve y = x^6 at the point (2, 0)?
0
Study Notes
Application of Derivatives
Equations of Tangents
- The equation of the tangent to the curve y = e^(-|x|) at the point where the curve cuts the line x = 1 is one of the options (e^(x+y) = 1, y = e^x, x + y = e, or none of these).
- The equation of the tangent to the curve y = e^(2x) + x^2 at x = 0 is related to the distance between the origin and the normal to the curve.
Tangents and Intersections
- The number of tangents that can be drawn from (2, 0) to the curve y = x^6 is one of the options (0, 1, 2, or 3).
- The number of tangents to the curve y^2 - 2x^3 - 4y + 8 = 0 that pass through (1, 2) is one of the options (1, 2, 3, or 6).
- The curves y^2 = 16x and y^3 = 16x intersect at right angles if 3a^2 is equal to one of the options (1, 2, 3, or 4).
Intersection and Angles
- If the curves y = 1 - ax^2 and y = x^2 intersect orthogonally, then a is one of the options (1, 1/2, 2, or 3).
- If the tangent at any point P(4m^2, 8m^3) of the curve x^3 - y^2 = 0 is a normal also to the curve x^3 - y^2 = 0, then find the value of (9m^2 + 2).
- If the curves y = ax and y = e^x intersect at an angle α, then tan α equals one of the options (loge(a)/(1 + loge(a)), (1 + loge(a))/(1 - loge(a)), loge(a) - 1, or none of these).
Other Applications
- The area of a triangle formed by a tangent to the curve 2xy = a^2 and the coordinate axes is one of the options (2a^2, a^2, 3a^2, or none of these).
- The normal of the curve x = a(cos θ + θ sin θ), y = a(sin θ - θ cos θ) at any θ is such that it makes a constant angle with x-axis, passes through the origin, or is at a constant distance from the origin, or none of these.
Solve problems involving equations of tangents and intersections of curves using derivatives. Topics include finding equations of tangents and determining the number of tangents that can be drawn to a curve.
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