If the coordinates of point, of contact of the circles $\mathrm{x}^2+\mathrm{y}^2-4 \mathrm{x}+8 \mathrm{y}+4=0$ and $\mathrm{x}^2+\mathrm{y}^2+2 \mathrm{x}=0$ is $(\mathrm{a}, \mathrm{b})$, then $a+2 b=$

• A

  -1
• Correct answer
  B

  -2
• C

  0
• D

  1

Solution

Circle $x^2+y^2-4 x+8 y+4=0$ have centre $P(2,-4)$ and radius 4 unit and circle $x^2+y^2$ $+2 x=0$ have centre $Q(-1,0)$ and radius 1 unit
Let $O(a, b)$ be the point of contact. $$ \begin{aligned} & O(a, b)=\frac{4(-1)+1(2)}{4+1}, \frac{4(0)+1(-4)}{4+1} \\ & =\left(-\frac{2}{5},-\frac{4}{5}\right) \\ & \therefore a+2 b=-2 \end{aligned} $$

Hence, the correct answer is option: 'B'.