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. 0.0
• B. 1
• C. 2
• D. Infinite

Answer 1

The correct option is A 0.0

The equation of line having slope m and passing through $(4,-5)$ is
$y+5=m(x-4)$
$\Rightarrow mx-y-(4m+5)=0$
The distance of a point $(-2,3)$ from the line $mx-y-(4m+5)=0$ is 12
So, $\left|\frac{m(-2)-3-(4m+5)}{\sqrt{m^2+1}}\right|=12$
$\Rightarrow 36m^2+64+96m=144m^2+144$
$\Rightarrow 108m^2-96m+80=0$
$\Rightarrow 27m^2-24m+20=0$
Now, $b^2-4ac=(-24{)}^2-4\times 27\times 20=-1584$
Thus, $b^2-4ac<0$
So slope can not be imaginary number
$\therefore$ No line can be drawn through the point $(-2,3)$.