Question

The focal chord to $y^2=16x$ is tangent to $(x-6{)}^2+y^2=2$, then the possible values of the slope of this chord, are

• A. $\{-1,1\}$
• B. $\{-2,2\}$
• C. $\{-2,\frac{1}{2}\}$
• D. $\{-2,-\frac{1}{2}\}$

Answer 1

The correct option is A $\{-1,1\}$

Here, the focal chord of $y^2=16x$ is tangent to circle ${(x-6)}^2+y^2=2$

$\Rightarrow$ focus of parabola as $(a,0)$ i.e $(4,0)$

$\Rightarrow$ centre and radius of circle is$(6,0)$ and $\sqrt{2}$ respectively

Thus the equation of focal chord will be

$(y-0)=m(x-4)$

$mx-y-4m=0$

As this line is tangent to the circle

$\therefore \frac{|6m-4m|}{\sqrt{m^2+1}}=\sqrt{2}$

$\Rightarrow \frac{4m^2}{m^2+1}=2$

$4m^2=2m^2+2$

$2m^2=2\Rightarrow m^2=1$

$m=\pm 1$

therefore the slope of the focal chord as a tangent to circle $=\pm 1$