Question

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

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

Answer 1

The correct option is A $0$

Equation of line with slope $m$ passing through point $(4,-5)$ is given by,

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

Given distance of above line from the point $(-2,3)$ is $12$

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

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

$\Rightarrow (3m+4{)}^2=36(m^2+1)$, square both sides

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

Discriminant of of above quadratic is, $(-24{)}^2-4\cdot 27\cdot 20=-1584<0$

Hence, there no real root of above quadratic.

i.e. there does not exist any line at a distance $12$ from the point $(-2,3)$

and passing through the point $(4,-5)$