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10-1.Circle and System of Circles
hard
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?
A
$3 x-4 y-24=0$
B
$3 x+4 y-6=0$
C
$4 x+3 y-8=0$
D
$4 x-3 y+17=0$
(JEE MAIN-2020)
Solution
Equation of family of circle touching y-axis at $(0,4)$ is given by $(x-0)^{2}+(y-4)^{2}+\lambda x=0$
It passes through $(2,0)$ $\Rightarrow \quad \lambda=-10$
$\Rightarrow \quad$ Required circle is $(x-0)^{2}+(y-4)^{2}-10 x=0$
$\Rightarrow \quad x^{2}+y^{2}-10 x-8 y+16=0$
center of circle $\equiv(5,4)$ and radius $=5$ distance of $4 \mathrm{x}+3 \mathrm{y}-8=0$ from $(5,4)$
$=\left|\frac{24}{5}\right| \neq$ radius
Standard 11
Mathematics
