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
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
Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.
Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
Find a point on the curve y = (x – 2)2. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
The points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
