Question

Equation of the common tangent to $y^2=8x$ and $3x^2-y^2=3$ is

• A. $2x+y+1=0$
• B. $x+2y-5=0$
• C. $2x+y-6=0$
• D. $2x-y+7=0$

Answer 1

Answer: (A)

$2x+y+1=0$

Equation of tangent to the parabola $y^2=8x$ is
$y=mx+\frac{a}{m}=mx+\frac{2}{m}...\left(1\right),$ (here $a=2)$
Given hyperbola is, $\frac{x^2}{1}-\frac{y^2}{3}=1$
$\Rightarrow a^2=1,b^2=3$
Also line (1) is also tangent to the hyperbola, so using condition of tangency, $c^2=a^2{m}^2-b^2$
$\Rightarrow \frac{4}{m^2}=m^2-3$
$\Rightarrow m^2=4,m^2=-1$ (not possible)
$\Rightarrow m=\pm 2$
Therefore, common tangent is $2x\pm y+1=0$
Hence, option 'A' is correct.