# 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 - Mathematics

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 costcos3t

y=3 sintsin3t

Slope of the tangentdy/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.

Concept: Tangents and Normals
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