QJEEM22Ma35d3: JEE Main 2022 Mathematics - Class 11 JEEM22Ma35d3 Solution: In an isosceles triangle ABC, the vertex A is (6, 1) and the…

$$\begin{array}{c}2x+y=4\hfill \\ 2x+6y=14\hfill \end{array}\}y=2,x=3$$

B(1, 2)

Let C(k, 4 $-$ 2k)

Now $A{B^2} = A{C^2}$

${5^2} + {( - 1)^2} = {(6 - k)^2} + {( - 3 + 2k)^2}$

$\Rightarrow 5{k^2} - 24k + 19 = 0$

$(5k - 19)(k - 1) = 0 \Rightarrow k = {{19} \over 5}$

$C\left( {{{19} \over 5}, - {{18} \over 5}} \right)$

Centroid ( $\alpha$ , $\beta$ )

$\alpha = {{6 + 1 + {{19} \over 5}} \over 3} = {{18} \over 5}$

$\beta = {{1 + 2 - {{18} \over 5}} \over 3} = - {1 \over 5}$

Now $15(\alpha + \beta )$

$15\left( {{{17} \over 5}} \right) = 51$