Question

The number of lines that can be drawn through the point $(4,-5)$ at distance of 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 line through Point$(4,-5)$ with slope m

$y+5=mx-4m$

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

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

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

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

$\left|\frac{-3m-4}{\sqrt{m^2+1}}\right|=6$

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

On squaring both sides

$\frac{9m^2+16+24m}{m^2+1}=36$

$9m^2+16+24m=36m^2+36$

$27m^2-24m+20=0$

$m=\frac{24\pm \sqrt{576-2160}}{54}$

$m=\frac{24\pm \sqrt{-1584}}{54}$

$m$ has imaginary roots

Hence No such line is possible