# 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 x2 + y2 = 2a2 and y2 = 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 x2 + y2 = 2a2 is

xcosθ + ysinθ = √2a . . . . . (1)

The equation of any tangent to the parabola y2 = 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, m4 + m2 – 2 = 0

⇒ m2 = 1 ⇒ m = ± 1

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

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