Question

If $x+2y+k=0$ is a common tangent to $y^2=4x\&\frac{x^2}{a^2}+\frac{y^2}{3}=1(a>\sqrt{3}).$ then the value of a, k and other common tangent are given by :

• A. a = 2
• B. a = -2
• C. x + 2y + 4 = 0
• D. k = 4

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A k = 4
B a = -2
C a = 2
D x + 2y + 4 = 0

Given slope of common tangent $m=\frac{1}{2}$
Equation of general tangent to $y^2=4x$ is
$y=mx+\frac{1}{m}$ ..... (i)
$\Rightarrow y=\frac{1}{2}x+2$

$m=\frac{1}{2}$ in given equation

On comparing with given equation, we get $k=4$

Equation of tangent of $\frac{x^2}{a^2}+\frac{y^2}{3}=1$ is
$y=m\pm \sqrt{a^2{m}^2+3}$ .....(ii)
On comparing (i) & (ii)
$\frac{1}{m}=\pm \sqrt{a^2{m}^2+3}$ .....(iii)
$\Rightarrow a^2=4\Rightarrow a=\pm 2$ ........(iv)
Using (iii) & (iv) we get $m=\pm \frac{1}{2}$.
So equation of other common tangent is $x+2y+4=0$