Question

Find the equations of the straight lines touching both $x^2+y^2=2$ and $y^2=8x.$

Answer 1

For the parabola $y^2=8x$

$a=2$

Equation of tangent in slope form for parabola

$y=mx+\frac{2}{m}$

$y-mx-\frac{2}{m}=0$

We know, if this is the common tangent than it will pass through circle hence its distance from the centre of circle will be its radius. Then,

$\frac{|0-0-\frac{2}{m}|}{\sqrt{1^2+(-m{)}^2}}=5$

$2=m^2(1+m^2)$

$m^2+m^4-2=0$

$(m^2+2)(m^2-1)=0$

$m^2-1=0$

$m=\pm 1$

Hence, the required tangents are $x\pm y+2=0$.