Two vertices of a triangle are and , and its orthocentre is . If the coordinates of the point are and the centre of the of the circle circumscribing the triangle is , then the value of equals

Let the circumcentre be Solving (i) and (ii)

Two vertices of a triangle are and , and its orthocentre is... | JEE Main 2024 Properties of Triangle

Q: JEE Main 2024 Properties of Triangle: Two vertices of a triangle are and , and its orthocentre is . If the coordinates of the point are an...

The correct answer is A. Let the circumcentre be Solving (i) and (ii)

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Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$ , and its orthocentre is $\mathrm{P}(1,1)$ . If the coordinates of the point $\mathrm{C}$ are $(\alpha, \beta)$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$ , then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals

$\begin{aligned} & m_{P A}=\frac{2}{-2}=-1 \\ & \therefore \quad m_{B C}=1 \end{aligned}$ $\begin{aligned} & B C: y=x+5 \\ & m_{B P}=\frac{2}{-3}=\frac{-2}{3} \\ & \therefore m_{A C}=\frac{3}{2} \\ & A C: y=\frac{3}{2} x-\frac{11}{2} \quad \Rightarrow 2 y=3 x-11 \\ & \therefore \quad C:(21,26) \end{aligned}$ Let the circumcentre be $(h, k)$ $\begin{aligned} & (h-1)^2+(k-1)^2=(h+2)^2+(k-3)^2 \quad \text{... (i)}\\ & (h-1)^2+(k-1)^2=(h-3)^2+(k+1)^2 \quad \text{... (ii)} \end{aligned}$ Solving (i) and (ii) $\begin{aligned} & h=\frac{-19}{2}, k=\frac{-23}{2} \\ & \alpha+\beta+2(h+k) \\ & =21+26-19-23 \\ & =2+3=5 \end{aligned}$

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