Question

A tangent to the parabola sq(y)=4ax at P(p,q) is perpendicular to tangent at other point Then the coordinates of Q are

Answer 1

if tangents are pperpendicular then tchord formed by it is a focal chord .use formula os relation between end pts of focal chord then u will get 'a' option correct 

Answer 2

The coordinates of Q are  $(\frac{a^{2} }{p}, - \frac{4a^{2} }{q})$.

Step-by-step explanation:

Given the equation of the parabola y² = 4ax

Now, differentiating with respect to x, we get

$2y\frac{dy}{dx} = 4a$

⇒ $\frac{dy}{dx} = \frac{2a}{y}$

Therefore, the slope of the tangent at point P(p,q) will be $\frac{2a}{q}$

Again, the slope of the tangent at the required point Q(h,k) will be $\frac{2a}{k}$.

Since those two lines are perpendicular to each other, hence $(\frac{2a}{q})(\frac{2a}{k}) = - 1$

⇒ $k = - \frac{4a^{2}}{q}$

Now, we have $k^{2} = 4ah$

⇒ $\frac{16a^{4}}{q^{2}} = 4ah$

⇒ $h = \frac{4a^{3}}{q^{2}}$

Now, we have q² = 4ap

Therefore, $h = \frac{4a^{3} }{4ap} = \frac{a^{2} }{p}$

Hence, (h,k) ≡ $(\frac{a^{2} }{p}, - \frac{4a^{2} }{q})$ (Answer)