Question

The number of lines which pass through the point $(2,-3)$ and are at the distance 8 from the point $(-1,2)$ is

• A. Infinite
• B. $4$
• C. $2$
• D. $0$

Answer 1

The correct option is D $0$

Equation of line from point $(2,-3)$

$y+3=m(x-2)$

$y+3=mx-2m$

$mx-y-2m-3=0$

The distance of above from point $(-1,2)$ is 8

$\left|\frac{-m-2-2m-3}{\sqrt{m^2+1}}\right|=8$

$\left|\frac{-3m-5}{\sqrt{m^2+1}}\right|=8$

$\frac{3m+5}{\sqrt{m^2+1}}=8$

$3m+5=8\sqrt{m^2+1}$

On squaring

$9m^2+25+30m=64m^2+64$

$55m^2-30m+39=0$

$b^2-4ac={30}^2-110\times 39$

$b^2-4ac=900-4290$

$b^2-4ac=-3390<0$

Hence roots are imaginary Hence no line