Planes in 3D

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Question

If $5 a + 5 b + 20 c = t,$ then the value of t for which the line $ax + by + c - 1 = 0$ always passes through a fixed point is

Moderate

Solution

Equation of line $\frac{ax}{c - 1} + \frac{by}{c - 1} + 1 = 0$ has two independent parameters.
$5 a + 5 b + 20 c = t$
$∴ 5 a + 5 b = t - 20 c$
$\frac{5 a}{c - 1} + \frac{5 b}{c - 1} = \frac{t - 20 c}{c - 1}$
R.H.S is independent of c if $t = 20 .$

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