## The equation of the straight line(s) that touches both x^2 + y^2 = 2a^2 and y^2 = 8ax is /are

Q: The equation of the straight line(s) that touches both x^{2} + y^{2} = 2a^{2} and y^{2} = 8ax is /are

(A) y = x + 2a

(B) y = – x – 2a

(C) y = – x + 2a

(D) y = x – 2a

Sol. The equation of any tangent to the circle x^{2} + y^{2} = 2a^{2} is

xcosθ + ysinθ = √2a . . . . . (1)

The equation of any tangent to the parabola y^{2} = 8ax is

y = mx + 2a/m . . . . . (2)

Since (1) and (2) are identical

$\large \frac{cos\theta}{-m} = \frac{sin\theta}{1} = \frac{m}{\sqrt{2}}$

$\large cos\theta = \frac{-m^2}{\sqrt{2}} $ and $\large sin\theta = \frac{m}{\sqrt{2}} $

Squaring and adding, m^{4} + m^{2} – 2 = 0

⇒ m^{2} = 1 ⇒ m = ± 1

Substituting in (2) the equation of the required tangent is y = ± (x + 2a)

Hence (A) and (B) are the correct answers.