Let the point B be the reflection of the point $\mathrm{A}(2,3)$ with respect to the line $8 \mathrm{x}-6 \mathrm{y}-23=0$. Let $\Gamma_{\mathrm{A}}$ and $\Gamma_{\mathrm{B}}$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles $\Gamma_{\mathrm{A}}$ and $\Gamma_{\mathrm{B}}$ 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 ____

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Circle

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

Γ_{A}

and

Γ_{B}

be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles

Γ_{A}

and

Γ_{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 ____

Answer: 10

Solution

The distance of point A from the given line is

\frac{∣8 ( 2 ) - 6 ( 3 ) - 23∣}{64 + 36} = \frac{5}{2}

.

Since B is the reflection of A, the distance

A B = 2 \times \frac{5}{2} = 5

.

For the common external tangent, by similar triangles:

\frac{CA}{CB} = \frac{2}{1}

Since C, A, B are collinear:

\frac{AC}{AB} = \frac{2}{1}

Therefore:

AC = 2 \times 5 = 10

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Numerical answer

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JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Circle

Solution

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