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

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (C)

$2$

Since the angle $RPQ$ is a right angle,
$\therefore$ Slope of $RP\times$ slope of $PQ=-1$
$\Rightarrow \frac{1-y}{3-x}\times \frac{5-1}{6-3}=-1\Rightarrow 3x+4y=13$ ...(1)
Also, area of $△RPQ=7\Rightarrow \frac{1}{2}\left|\begin{array}{ccc}x& y& 1\\ 3& 1& 1\\ 6& 5& 1\end{array}\right|=7$
$\Rightarrow \frac{1}{2}[x(1-5)-y(3-6)+1(15-6)]=\pm 7$
$\Rightarrow -4x+3y+9=\pm 14\Rightarrow -4x+3y=5$ and $-4x+3y=-23$

Solving equation of these two lines with (1), we get two different coordinates of the point $R$.
So, there are two such points $R$.