Question

Which of the following point lies on the common tangent at the circle $x^2+y^2-4x-6y-36=0$ and $x^2+y^2-10x+2y+22=0$.

• A. $(10,\frac{1}{4})$
• B. $(\frac{1}{4},10)$
• C. $(12,2)$
• D. $(2,12)$

Answer 1

The correct option is A $(10,\frac{1}{4})$

$C_1:(x-2{)}^2+(y-3{)}^2=7^2,c_1:(2,3)C_2:(x-5{)}^2+(y+1{)}^2=2^2,c_2:(5,-1)$

$c_1{c}_2=\sqrt{(2-5{)}^2+(3+1{)}^2}=5\Rightarrow c_1{c}_2=r_1-r_2=7-2=5$


Therefore $C_2$ lies inside the $C_1$ and touches only at one point.
Therefore, equation of tangent is,
$C_1-C_2=0\Rightarrow 3x-4y=29$
From the given options
$(10,\frac{1}{4})$ lies on the common tangent.