Question

If a circle, whose centre is $(-1,1)$ touches the straight line $x+2y$ $+12=0$, then the coordinates of the point of contact are

• A. $\left(\frac{-7}{2},-4\right)$
• B. $\left(\frac{-18}{5},\frac{-21}{5}\right)$
• C. $(2,-7)$
• D. $(-2,-5)$

Answer 1

Answer: (B)

Let point of contact be $P\left(x_1,y_1\right)$.
This point lies on line
$x_1+2y_1=-12$ ... (i)
Gradient of $PO=m_1=\frac{y_1-1}{x_1+1}$
Gradient of $x+2y+12=m_2=-\frac{1}{2}$
The two lines are perpendicular
$\therefore m_1{m}_2=-1$
$\Rightarrow \left(\frac{y_1-1}{x_1+1}\right)\left(\frac{-1}{2}\right)=-1$
$\Rightarrow y_1-1=2x_1+2$
$\Rightarrow 2x_1-y_1=-3$ ... (ii)
On solving equation (i) and (ii), we get
$\left(x_1,y_1\right)=\left(\frac{-18}{5},\frac{-21}{5}\right)$