Q86 / 160 · TS EAMCET 2024 (10 May Shift 2) · Mathematics

If the line $2 x+b y+5=0$ forms an equilateral triangle with $a x^2-96 b x y+k y^2=0$, then $a+3 k=$

A
$3 b$
Correct answer
B
192
C
$4 b^2$
D
102

Solution

We know that pair of lines $\left(a^2-3 b^2\right) x^2+8 a b x y+\left(b^2-3 a^2\right) y^2=0$ with the lines $a x+b y+c=0$ forms an equilateral triangle. So, put $a=2, c=5$ $\left(4-3 b^2\right) x^2+16 b x y+\left(b^2-12\right) y^2=0$...(i) Given pair of lines : $a x^2-96 b x y+k y^2=0$ $\Rightarrow 2 x^2-96 b x y+1 y^2=0 \Rightarrow-\frac{x}{3}+16 b x y-\frac{k}{6} y=0$...(ii) Comparing (i) and (ii) $\begin{aligned} & 4-3 b^2=\frac{-1}{3} ; 12-9 b^2=-1 ; b^2=\frac{13}{9} \\ & \text { and } b^2-12=\frac{-k}{6} ; \frac{13}{9}-12=\frac{-k}{6} \Rightarrow k=\frac{190}{3} \\ & \text { then } a+3 k=192\end{aligned}$

Hence, the correct answer is option: 'B'.

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