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10-1.Circle and System of Circles
easy
Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8 x-6 y-23=0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is. . . . . .
A
$10$
B
$15$
C
$20$
D
$25$
(IIT-2019)
Solution
Distance of point $A$ from given line $=\frac{5}{2}$
$\frac{ CA }{ CB }=\frac{2}{1}$
$\Rightarrow \frac{ AC }{ AB }=\frac{2}{1}$
$\Rightarrow AC =2 \times 5=10$
Standard 11
Mathematics
