The normal at a point θ to the curve x=a(1+ cos θ), y=a sin θ always passes through the fixed point

Question

The normal at a point θ to the curve x=a(1+ cos θ), y=a sin θ always passes through the fixed point (A) (0, a) (B) (2 a, 0) (C) (a, 0) (D) (a, a)

Tardigrade Admin · Accepted Answer

Given curve is x=a(1+ cos θ), y=a sin θ ⇒ (x-a)2+y2=a2, is a circle. The normal to the circle always passes through the centre of the circle. So, normal at a point to the given curve passes through centre (a, 0)

Q. The normal at a point $θ$ to the curve $x = a (1 + cos θ), y = a sin θ$ always passes through the fixed point

$(0, a)$

$(2 a, 0)$

$(a, 0)$

$(a, a)$

Given curve is $x = a (1 + cos θ), y = a sin θ$
$\Rightarrow (x - a)^{2} + y^{2} = a^{2}$ , is a circle.
The normal to the circle always passes through the centre of the circle.
So, normal at a point to the given curve passes through centre $(a, 0)$

Q. The normal at a point θ to the curve x=a(1+cosθ),y=asinθ always passes through the fixed point

(0,a)

(2a,0)

(a,0)

(a,a)