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Perpendicular Form of a Straight Line

If the line's...

Question

If the line's $2 x + 3 y - 5 = 0$ and $(t^{2} + 1) x + (2 t - 3) y + 5 = 0$ are perpendicular to each other, then absolute value of sum of all possible real values of $t$ is

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Solution

Given lines
$l_{1} : 2 x + 3 y - 5 = 0 l_{2} : (t^{2} + 1) x + (2 t - 3) y + 5 = 0$
Slope of both the lines is
$m_{1} = - \frac{2}{3}, m_{2} = - \frac{t^{2} + 1}{2 t - 3}$
So,
$m_{1} \times m_{2} = - 1 \Rightarrow \frac{2}{3} \times (\frac{t^{2} + 1}{2 t - 3}) = - 1 \Rightarrow 2 t^{2} + 2 = 9 - 6 t$
$\Rightarrow 2 t^{2} + 6 t - 7 = 0$
$\Rightarrow t_{1} + t_{2} = - 3$
$∴ | t_{1} + t_{2} | = 3$

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