Question

Show that the normal at any point theta to the curve x=a(cos theta+ theta sin theta),y=a(sin theta-theta cos theta) is at a constant distance from the origin

Answer 1

Answer:

Step-by-step explanation:

Toolbox:

ddx(sinθ)=cosθ

ddx(cosθ)=−sinθ

ddx(xn)=nxn−1

Step 1:

We have x=acosθ+aθsinθ

y=asinθ−aθsinθ

Differentiating with respect to x

dxdθ=−asinθ+aθcosθ+asinθ

=aθcosθ

dydθ=acosθ−acosθ+aθsinθ

=aθsinθ

Step 2:

Slope of the tangent =dydx=dydθ÷dxdθ

=aθsinθaθcosθ

=sinθcosθ

=tanθ

Step 3:

Slope of the normal at θ=−1dydx

⇒−1tanθ

⇒−cotθ

The equation of the normal at the point ′θ′ is

[y−(asinθ−aθcosθ)]=−cotθ[x−acosθ+aθsinθ]

On simplifying,we get

xcosθ+ysinθ=a which is the equation of the normal.