JEE MAIN - Mathematics (2023 - 15th April Morning Shift - No. 11)
If $(\alpha, \beta)$ is the orthocenter of the triangle $\mathrm{ABC}$ with vertices $A(3,-7), B(-1,2)$ and $C(4,5)$, then $9 \alpha-6 \beta+60$ is equal to :
30
40
25
35
Explanation
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$$ \begin{aligned} & \text { Altitude of BC: } y+7=\frac{-5}{3}(x-3) \\\\ & 3 y+21=-5 x+15 \\\\ & 5 x+3 y+6=0 \\\\ & \text { Altitude of AC: } y-2=\frac{-1}{12}(x+1) \\\\ & 12 y-24=-x-1 \\\\ & x+12 y=23 \\\\ & \alpha=\frac{-47}{19}, \quad \beta=\frac{121}{57} \\\\ & 9 \alpha-6 \beta+60=25 \end{aligned} $$
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