First slide

Cartesian plane

Question

$If A (5, 2), B (10, 12) and P (x, y) is such that \frac{A P}{P B} = \frac{3}{2}, then the internal bisector of ∠ A P B always passes through$

Easy

A

(20,32)
B

(8,8)
C

(8, -8)
D

(-8, -8)

Solution

$Let internal bisector of ∠ A P B meets the side A B at C .$
$\begin{array}{l} We know that \frac{A C}{B C} = \frac{A P}{P B} \\ ∴ \frac{A C}{B C} = \frac{3}{2} \\ ∴ coordinates of C are (\frac{3 (10) + 2 (5)}{3 + 2}, \frac{3 (12) + 2 (2)}{3 + 2}) or (8, 8) \end{array}$
$Hence the internal bisector of ∠ A P B always passes through$ $point (8, 8)$

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