Question

For the circles $x^2+y^2-8x+6y+21=0$ and $x^2+y^2+4x-10y-115=0,$ which of the following statements is/are true?

• A. Number of common tangents is $3$
• B. Number of common tangent is $1$
• C. The point of contact is $P=(5.2,-4.6)$
• D. The point of contact is $P=(-5.2,4.6)$

Answer 1

Answer: (B), (C)

The correct options are
B Number of common tangent is $1$
C The point of contact is $P=(5.2,-4.6)$

For circle $1:$
$C_1(4,-3)$ and $r_1=2$
For circle $2:$
$C_2(-2,5)$ and $r_2=12$

Now, $C_1{C}_2=\sqrt{36+64}=10$
$\Rightarrow C_1{C}_2=|r_1-r_2|$
So, they touch internally.
Point of contact $P$ divides $C_1{C}_2$ in external ratio of $r_1:r_2$, so
$P=(\frac{48+4}{12-2},\frac{-36-10}{12-2})\therefore P=(5.2,-4.6)$