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QJEEM24Madc0f: JEE Main 2024 Mathematics - Class 11 JEEM24Madc0f Solution: If the orthocentre of the triangle formed by the lines and , is …

If the orthocentre of the triangle formed by the lines 2x+3y−1=0,x+2y−1=02 x+3 y-1=0, x+2 y-1=0 and ax+by−1=0a x+b y-1=0 , is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3,4)(3,4) and (−6,−8)(-6,-8) , then the value of ∣a−b∣|a-b| is _________.

Step-by-Step Verified Solution

Let circumcentre, orthocentre and centroid of a triangle PQRP Q R are C1,H1C_1, H_1 and G1G_1 respectively

JEE Main 2024 (Online) 8th April Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 15 English Explanation 1

⇒G1≡(0,0)\Rightarrow G_1 \equiv(0,0) orthocentre of △ABC\triangle A B C is (0,0)(0,0)

JEE Main 2024 (Online) 8th April Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 15 English Explanation 2

amp;mAH2=+ba⇒a+b=0amp; eq n  of lines H2C is y=32xamp;⇒ point C≡(14,38) lies on ax+by−1=0amp;⇒a4+38b−1=0⇒a4−38a−1=0amp;⇒a=−8,b=8amp;∣a−b∣=16\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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