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Questions and Answers
If the curve y = 1 – ax2 and y = x2 intersect orthogonally, then a is equal to
If the curve y = 1 – ax2 and y = x2 intersect orthogonally, then a is equal to
- 1/3
- 1/12 (correct)
- 1/2
- 1/4
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
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 makes a constant angle with the y-axis
- It is at a constant distance from the origin (correct)
- It passes through the origin
- It makes a constant angle with the x-axis
The area of a triangle formed by a tangent to the curve 2xy = a2 and the coordinate axes is
The area of a triangle formed by a tangent to the curve 2xy = a2 and the coordinate axes is
- 2a2
- a2
- 3a2 (correct)
- None of these
If the curves y = a x and y = ex intersect at an angle α, then tan α equals
If the curves y = a x and y = ex intersect at an angle α, then tan α equals
What is the condition for the curves y = 1 – ax2 and y = x2 to intersect orthogonally?
What is the condition for the curves y = 1 – ax2 and y = x2 to intersect orthogonally?
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?
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?
What is the area of a triangle formed by a tangent to the curve 2xy = a2 and the coordinate axes?
What is the area of a triangle formed by a tangent to the curve 2xy = a2 and the coordinate axes?
If the curves y = a x and y = ex intersect at an angle α, what is the value of tan α?
If the curves y = a x and y = ex intersect at an angle α, what is the value of tan α?
What is the equation of the tangent to the curve y = e^–|x| at the point where the curve cuts the line x = 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?
What is the distance between the origin and the normal to the curve y = e^(2x) + x^2 at x = 0?
What is the distance between the origin and the normal to the curve y = e^(2x) + x^2 at x = 0?
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?
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?
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)?
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)?
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?
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?
What is the equation of the curve where the tangent at A except (0, 0) meets the curve again at B?
What is the equation of the curve where the tangent at A except (0, 0) meets the curve again at B?
How many tangents can be drawn from the point (2, 0) to the curve y = x^6?
How many tangents can be drawn from the point (2, 0) to the curve y = x^6?
What is the equation of the curve in parametric form, given x = sec^2 t and y = cot t?
What is the equation of the curve in parametric form, given x = sec^2 t and y = cot t?
How many tangents to the curve y^2 – 2x^3 – 4y + 8 = 0 pass through the point (1, 2)?
How many tangents to the curve y^2 – 2x^3 – 4y + 8 = 0 pass through the point (1, 2)?
If the curves x^2 + y^2 = 1 and y^3 = 16x intersect at right angles, what is the value of 3a^2?
If the curves x^2 + y^2 = 1 and y^3 = 16x intersect at right angles, what is the value of 3a^2?
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?
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?
How many points of intersection are there between the curves x^2 + y^2 = 1 and y^3 = 16x?
How many points of intersection are there between the curves x^2 + y^2 = 1 and y^3 = 16x?
What is the equation of the tangent to the curve y = x^6 at the point (2, 0)?
What is the equation of the tangent to the curve y = x^6 at the point (2, 0)?
What is the slope of the tangent to the curve y = x^6 at the point (2, 0)?
What is the slope of the tangent to the curve y = x^6 at the point (2, 0)?
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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.
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