JEE MAIN - Mathematics (2024 - 8th April Morning Shift - No. 21)
If the orthocentre of the triangle formed by the lines $$2 x+3 y-1=0, x+2 y-1=0$$ and $$a x+b y-1=0$$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $$(3,4)$$ and $$(-6,-8)$$, then the value of $$|a-b|$$ is _________.
Answer
16
Explanation
Let circumcentre, orthocentre and centroid of a triangle $$P Q R$$ are $$C_1, H_1$$ and $$G_1$$ respectively
_8th_April_Morning_Shift_en_21_1.png)
$$\Rightarrow G_1 \equiv(0,0)$$ orthocentre of $$\triangle A B C$$ is $$(0,0)$$
_8th_April_Morning_Shift_en_21_2.png)
$$\begin{aligned} & m_{A H_2}=+\frac{b}{a} \Rightarrow a+b=0 \\ & \text { eq }{ }^{\text {n }} \text { of lines } H_2 C \text { is } y=\frac{3}{2} x \\ & \Rightarrow \text { point } C \equiv\left(\frac{1}{4}, \frac{3}{8}\right) \text { lies on } a x+b y-1=0 \\ & \Rightarrow \frac{a}{4}+\frac{3}{8} b-1=0 \Rightarrow \frac{a}{4}-\frac{3}{8} a-1=0 \\ & \Rightarrow a=-8, b=8 \\ & |a-b|=16 \end{aligned}$$
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