Question:
A circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to this circle?
Correct Option: , 4
Solution:
Equation of family of circle
$(x-0)^{2}+(y-4)^{2}+\lambda x=0$
Passes through the point $(2,0)$ then
$4+16+2 \lambda=0 \Rightarrow \lambda=-10$
Hence, the equation of circle
$x^{2}+y^{2}-10 x-8 y+16=0$
$\Rightarrow(x-5)^{2}+(y-4)^{2}=25$
Centre $(5,4)$.
$R=\sqrt{\frac{1}{2} \text { coeff. of } x+\frac{1}{2} \text { coeff. of } y \text {-constant }}$
$=\sqrt{25+16-16}=5$
Perpendicular distance of $4 x+3 y-8=0$ from the centre of circle
$=\left|\frac{20+16-8}{\sqrt{16+9}}\right|=\frac{28}{5} \neq 5$
Hence, $4 x+3 y-8=0$ can not be tangent to the circle.