Exam

JEE Advanced

Year

2019

Paper

Paper 1

Subject

Mathematics

Chapter

Straight Lines

JEE Advanced 2019 · Paper 1 · Mathematics · Straight Lines

Let the point B be the reflection of the point A(2, 3) with respect...

Let the point $B$ be the reflection of the point $$A\left(2, 3\right)$ with respect to the line $8x - 6y - 23 = 0 .$ Let ${T}_{A} and${T}_{B}$be circle of radii$2$and$1$with centres$A$and$B$respectively. Let$T$be a common tangent to the circle${T}_{A}$and${T}_{B}$such that both the circle 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 ___________.

Step-by-step solution

Clearly,$ACP$ and $BCQ$ are similar triangles.
Hence,
$\dfrac{AC}{BC} = \dfrac{{r}_{1}}{{r}_{2}}$
$\dfrac{AC}{BC} = 2$
$AC = 2BC$ ...(i)

Now, since $B$ , is image of $A$ in line $8x - 6y = 23$
$AD = BD$
$AB = 2AD$ ...(ii)
$AD =$ distance of $A$ from line
$8x - 6y = 23$
$AD = \left|\dfrac{16 - 16 - 23}{10}\right| = \dfrac{5}{2}$
Now as $AC = 2BC$
$⇒ AC = 2 ( AC - AB )$
$⇒ AC = 2AB$
$⇒ AC = 4AD$ (from (ii))
$AC = 4 \times \dfrac{5}{2}$
$AC = 10$