JEE MAIN - Mathematics (2025 - 23rd January Morning Shift - No. 18)
Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c+2 d$ is equal to
$3$
$\frac{7}{3}$
$2$
$\frac{8}{3}$
Explanation
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$$\begin{aligned} & \text { Equation of lines } Q R=5 x+2 y+2=0 \\ & \text { Equation of lines } P R=10 x-3 y-38=0 \\ & \therefore \text { Point } R(2,-6) \\ & \text { Centroid }=\left(\frac{5-2+2}{3}, \frac{4+4-6}{3}\right) \\ & =\left(\frac{5}{3}, \frac{2}{3}\right) \\ & c+2 d=\frac{5}{3}+\frac{4}{3}=3 \end{aligned}$$
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