Question

Number of lines can be drawn through the point $(4,-5)$ so that its distance from $(-2,3)$ will be equal to $12$

• A. 0
• B. 1
• C. 2
• D. Infinite

Answer 1

The correct option is C 2

The general equation of a line is $ax+by+c=0$

Distance between the line and the point $(x_1,y_1)$ is given by

$\left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right|$

Let us find the number of lines passing through the point $(4,-5)$ whose slope is $m$

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

$\Rightarrow mx-y-4m-5=0$ ..$\left(1\right)$

Distance between the point $(-2,3)$ and the line $\left(1\right)$ is given by

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

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

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

$\Rightarrow -3m-4=6\left(\sqrt{1+m^2}\right)$ by squaring on both sides,we get

$\Rightarrow 9m^2+16+24m-36-36m^2=0$

$\Rightarrow 27m^2-24m-20=0$ is quadratic in $m$ where $a=27,b=-24,c=-20$

Discriminant$=\Delta =b^2-4ac={(-24)}^2-4\times 27\times -20={\left(24\right)}^2+4\times 27\times 20>0$

$\Rightarrow$ Roots are real and distinct

$\therefore$ there exists two roots $m_1$ and $m_2$ which are real and distinct

$\therefore$ there exists two different lines with different slopes.

Hence,number of lines $=2$