First slide

Cartesian plane

Question

P(3, 1), Q(6, 5) and R(x, y) are three points such that the angle RPQ is a right angle and the area of $∆$ RPQ =
7, then the number of such points R is

Moderate

A

0
B

1
C

2
D

4

Solution

Since the angle RPQ is a right angle,
$∴$ slope of RP × slope of PQ = –1
$\Rightarrow \frac{1 - y}{3 - x} \times \frac{5 - 1}{6 - 3} = - 1 \Rightarrow 3 x + 4 y = 13$ (1)
Also, area of $∆$ RPQ = 7
$\Rightarrow \frac{1}{2} |\begin{matrix} x & y & 1 \\ 3 & 1 & 1 \\ 6 & 5 & 1 \end{matrix}| = 7$
$\Rightarrow \frac{1}{2} [x (1 - 5) - y (3 - 6) + 1 (15 - 6)] = \pm 7$
⇒ – 4x + 3y + 9 = ± 14 ⇒ – 4x + 3y = 5 (2)
and – 4x + 3y = –23 (3)
Solving Eq. (1) and (2) and (1) and (3), we get two
different coordinates of the point R. So, there are two such points R.

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