Advertisements
Advertisements
प्रश्न
Find the slope of tangent to the curve x = sin θ and y = cos 2θ at θ = `pi/6`
उत्तर
Given, x = sin θ and y = cos 2θ
Differentiating w.r.t. θ, we get
`("d"x)/("d"theta)` = cos θ and `("d"y)/("d"theta)` = – 2sin 2θ
∴ `("d"y)/("d"x) = ((("d"y)/("d"theta)))/((("d"x)/("d"theta))) = (-2sin2theta)/(cos theta)`
Slope of the tangent at θ = `pi/6` is
`(("d"y)/("d"x))_(theta = pi/6) = (-2sin2(pi/6))/(cos(pi/6))`
= `(-2sin(pi/3))/(cos(pi/6))`
= `(-2 xx sqrt(3)/2)/(sqrt(3)/2)`
= – 2
APPEARS IN
संबंधित प्रश्न
Find the equations of tangents and normals to the following curves at the indicated points on them: `x^2 - sqrt(3)xy + 2y^2 = 5 "at" (sqrt(3),2)`
Find the equations of tangents and normals to the following curves at the indicated points on them : 2xy + π sin y = `2pi "at" (1, pi/2)`
Find the equations of tangents and normals to the following curves at the indicated points on them : x sin 2y = y cos 2x at `(pi/4, pi/2)`
Find the equations of tangents and normals to the following curve at the indicated points on them:
x = sin θ and y = cos 2θ at θ = `pi/(6)`
Find the equations of tangents and normals to the following curves at the indicated points on them : `x = sqrt(t), y = t - (1)/sqrt(t)` at = 4.
Find the point on the curve y = `sqrt(x - 3)` where the tangent is perpendicular to the line 6x + 3y – 5 = 0.
Find the points on the curve y = x3 – 2x2 – x where the tangents are parllel to 3x – y + 1 = 0.
Find the equations of the normals to the curve 3x2 – y2 = 8, which are parallel to the line x + 3y = 4.
If the line y = 4x – 5 touches the curves y2 = ax3 + b at the point (2, 3), find a and b.
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.
If each side of an equilateral triangle increases at the rate of `(sqrt(2)"cm")/sec`, find the rate of increase of its area when its side of length 3 cm.
Choose the correct option from the given alternatives:
Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.
Choose the correct option from the given alternatives :
If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x`, then ______.
Choose the correct option from the given alternatives :
The normal to the curve x2 + 2xy – 3y2 = 0 at (1, 1)
Choose the correct option from the given alternatives :
The equation of the tangent to the curve y = `1 - e^(x/2)` at the point of intersection with Y-axis is
Solve the following : If the curves ax2 + by2 = 1 and a'x2 + b'y2 = 1, intersect orthogonally, then prove that `(1)/a - (1)/b = (1)/a' - (1)/b'`.
Solve the following : Determine the area of the triangle formed by the tangent to the graph of the function y = 3 – x2 drawn at the point (1, 2) and the coordinate axes.
Solve the following : Find the equation of the tangent and normal drawn to the curve y4 – 4x4 – 6xy = 0 at the point M (1, 2).
The slope of the tangent to the curve x = 2 sin3θ, y = 3 cos3θ at θ = `pi/4` is ______.
The slope of the normal to the curve y = x2 + 2ex + 2 at (0, 4) is ______.
Find the slope of tangent to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`
Find the slope of normal to the curve 3x2 − y2 = 8 at the point (2, 2)
Find points on the curve given by y = x3 − 6x2 + x + 3, where the tangents are parallel to the line y = x + 5.
Find the equation of tangent to the curve y = 2x3 – x2 + 2 at `(1/2, 2)`.
Find the equation of the tangents to the curve x2 + y2 – 2x – 4y + 1 = 0 which is parallel to the X-axis.
The coordinates of the point on the curve y = 2x – x2, the tangent at which has slope 4 are ______.
If the line 2x – y + 7 = 0 touches the curve y = ax2 + bx + 5 at (1, 9) then the values of ‘a’ and ‘b’ are ______.
