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$\mathrm{P(3,1), Q(6,5) \: and \: R\left(x_1, y_1\right)}$ with Google Docs such that $\mathrm{\angle \mathrm{PRQ}=\frac{\pi}{2}}$ and area of $\mathrm{\triangle \mathrm{RPQ}=7}$ . Then the number of such points $\mathrm{R}$ is

Option: 1
0

Option: 2
1

Option: 3
2

Option: 4
4

Answers (1)

best_answer

As angle $\mathrm{ \mathrm{PRQ}=\frac{\pi}{2},}$

$\mathrm{\mathrm{PR} \perp \mathrm{RQ}}$

Or $\mathrm{\frac{y_1-1}{x_1-3} \frac{y_1-5}{x_1-6}=-1}$

Or $\mathrm{y_1^2+x_1^2-9 x_1-6 y_1+5+18=0}$
Also, area of $\mathrm{\triangle R P Q=7}$
$\Rightarrow \frac{1}{2}\left|\begin{array}{ccc} x_1 & y_1 & 1 \\ 3 & 1 & 1 \\ 6 & 5 & 1 \end{array}\right|=7 \text { or } 14=\mathrm{x}_1(-4)+\mathrm{y}_1(3)+(15-6)$
Or $\mathrm{4 x_1-3 y_1+5=0}$

From (1) and (2), we obtain

$\mathrm{ y_1^2+\left(\frac{3 y_1-5}{4}\right)^2-9 \frac{3 y_1-5}{4}-6 y_1+23=0 }$

Or $\mathrm{ 25 y_1^2-234 y_1+573=0 }$

Disc. $\mathrm{ =(234)^2-100 \times 573 }$

$\mathrm{ =54756-57300<0 }$

$\therefore$ No such point exists.

Hence option 1 is correct.

Posted by

Suraj Bhandari

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