If the orthocentre of the triangle formed by the lines and is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (-6, -8), then the value of is

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Solution

Q.
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 _________ . [2024]

Ans.
(16)

Let $∆ A B C$ be the given triangle with sides $2 x + 3 y = 1, x + 2 y = 1 and a x + b y = 1$

Let $∆ P Q R$ be another triangle whose circumcentre, orthocentre and centroid be $C_{1}, H_{1}$ and $G_{1}$ respectively. We know centroid divides circumcentre and orthocentre in the ratio 1 : 2.

$∴$ $Coordinates of G_{1} = (\frac{6 - 6}{3}, \frac{8 - 8}{3}) = (0, 0)$

$∴$ $Orthocentre of ∆ A B C, H_{2} = (0, 0)$

$Slope of A H_{2} = \frac{- 1}{Slope of B C} \Rightarrow \frac{1 - 0}{- 1 - 0} = \frac{- 1}{\frac{- a}{b}} \Rightarrow - a = b$

$Slope of A B = \frac{- 2}{3}$

Now, $slope of C H_{2} = \frac{3}{2}$

$∴$ $Equation of C H_{2} is given by y = \frac{3}{2} x$

Now, C is the point of intersection of $y = \frac{3}{2} x$ and $x + 2 y - 1 = 0$

$∴$ $C (\frac{1}{4}, \frac{3}{8}) are the coordinates of C which will also satisfy the equation of B C .$

$\Rightarrow \frac{a}{4} - \frac{3}{8} a - 1 = 0 \Rightarrow a = - 8, b = 8$

$\Rightarrow | a - b | = | - 8 - 8 | = 16$

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