If the tangents at the point P and Q on the circle x 2 + y 2 - 2 x + y = 5 meet at the point R ( 9 4 , ? 2 ) , then the area of the triangle PQR is [2023]

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Solution

Q.
If the tangents at the point P and Q on the circle $x^{2} + y^{2} - 2 x + y = 5$ meet at the point $R (\frac{9}{4}, 2)$ , then the area of the triangle PQR is [2023]

1 $\frac{13}{4}$

2 $\frac{5}{4}$

3 $\frac{5}{8}$

4 $\frac{13}{8}$

Ans.
(3)

Equation of circle(s) is, $x^{2} + y^{2} - 2 x + y - 5 = 0$

$\Rightarrow {(x - 1)}^{2} + {(y + \frac{1}{2})}^{2} = {(\frac{5}{2})}^{2}$

$\Rightarrow Radius (R) = \frac{5}{2}$

Now, length of tangent, $L = \sqrt{S_{1}} = \sqrt{\frac{81}{16} + 4 - 2 \times \frac{9}{4} + 2 - 5} = \frac{5}{4}$

So, area of $∆ P Q R$ = $\frac{R L^{3}}{R^{2} + L^{2}} = \frac{(\frac{5}{2}) {(\frac{5}{4})}^{3}}{{(\frac{5}{2})}^{2} + {(\frac{5}{4})}^{2}} = \frac{5}{8}$

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