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)

Given slope of common tangent $m=\frac{1}{2}$.
Equation of general tangent to $y^2=4x$ is
$y=mx+\frac{1}{m}\dots \dots \text{ (i) }$
$\Rightarrow y=\frac{1}{2}x+2\left[\because m=\frac{1}{2}\text{ in given equation }\right]$
On comparing with given equation, we get $k=4$
Equation of tangent of $\frac{x^2}{a^2}+\frac{y^2}{3}=1$ is
$y=mx\pm \sqrt{a^2{m}^2+3}$.....(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 $x+2y+4=0$