Advertisements
Advertisements
Question
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Advertisements
Solution
Given:
x=3 cost−cos3t
y=3 sint−sin3t
Slope of the tangent, `dy/dx=(dy/dt)/(dx/dt)=(3cost-3sin^2tcost)/(-3sint+3cos^2tsint)`
`=(3cost[cos^2t])/(-3sint[sin^2t])`
`dy/dx=(-cos^3t)/sin^3t`
∴Slope of the normal
`=sin^3t/cos^3 t`
The equation of the normal is given by
`(y-(3sint-sin^3t))/(x-(3cost-cos^3t))=sin^3t/cos^3t`
`=>ycos^3t-3sint cos^3t +sin^3tcos^3t=xsin^3t-3costsin^3t+sin^3tcos^3t`
`=>ycos^3t-xsin^3t=3(sintcos^3t-costsin^3t)`
`=>ycos^3t-xsin^3t=3sintcost(cos^2t-sin^2t)`
`=>ycos^3t-xsin^3t=3/2sin2tcos2t=3/4sin4t`
`=>4(ycos^3t-xsin^3t)=3sin4t`
Hence proved.
APPEARS IN
RELATED QUESTIONS
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Find the equations of the tangent and normal to the hyperbola `x^2/a^2 - y^2/b^2` at the point `(x_0, y_0)`
Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
Find the equation of the tangent and the normal to the following curve at the indicated point \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
The curve y = `x^(1/5)` has at (0, 0) ______.
At (0, 0) the curve y = x3 + x
The distance between the point (1, 1) and the tangent to the curve y = e2x + x2 drawn at the point x = 0
The number of common tangents to the circles x2 + y2 – 4x – 6x – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
Which of the following represent the slope of normal?
The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.
If m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points `(3cosθ, sqrt(3) sinθ)` and `(-3sinθ, sqrt(3) cos θ); θ ∈(0, π/2)`; then `(2 cot β)/(sin 2θ)` is equal to ______.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
