Question

The number of lines can be drawn through the point $(4,-5)$ at distance $12$ units from the point $(-2,3)$ is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is A $0$

Let equation of line from Point$(4,-5)$

$y+5=mx-4m\Rightarrow mx-y-(4m+5)=0$

Distance of above line from point $(-2,3)$ is $12$

$\left|\frac{-2m-3-4m-5}{m^2+1}\right|=12$

$\left|\frac{-6m-8}{m^2+1}\right|=12$

$\frac{6m+8}{m^2+1}=12$

$\frac{3m+4}{m^2+1}=6$

$3m+4=6m^2+6$

$\Rightarrow 6m^2-3m+2=0$

$m=\frac{3\pm \sqrt{9-48}}{12}$

$m=\frac{3\pm \sqrt{-39}}{12}$

Here the root of $m$ is imaginary SO $m$ has no real root

Hence Number of line Zero.