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10-1.Circle and System of Circles
hard
If the tangents at the points $P$ and $Q$ on the circle $x ^2+ y ^2-2 x + y =5$ meet at the point $R \left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is
A
$\frac{13}{4}$
B
$\frac{13}{8}$
C
$\frac{5}{4}$
D
$\frac{5}{8}$
(JEE MAIN-2023)
Solution
Equation of circle is $x^2+y^2-2 x+y-5=0$
$R=\frac{5}{2}$
Length of $P R=Q R=\sqrt{S_1}$
$=\sqrt{\frac{81}{16}+4-\frac{2 \times 9}{4}+2-5}=\frac{5}{4}$
Area of triangle $PQR =\frac{ RL ^3}{ R ^2+ L ^2}=\frac{\frac{5}{2} \cdot \frac{125}{64}}{\frac{25}{4}+\frac{25}{16}}=\frac{5}{8}$
Standard 11
Mathematics
