First slide

Equation of line in Straight lines

Question

If $9 a^{2} + 25 b^{2} - c^{2} + 30 a b = 0$ , then the set of lines $a x + b y + c = 0$ passes through the fixed point is

Moderate

A

$(3, 5)$
B

$(- 3, - 5)$
C

$(3, 5), (- 3, - 5)$
D

No fixed point

Solution

$The given condition is 9 a^{2} + 25 b^{2} - c^{2} + 30 a b = 0$
This can be rewrite as
$\begin{matrix} {(3 a + 5 b)}^{2} - c^{2} = 0 \\ (3 a + 5 b - c) (3 a + 5 b + c) = 0 \end{matrix}$
It implies that
$3 a + 5 b - c = 0, or 3 a + 5 b + c = 0$
$comparing the above equations with a x + b y + c = 0 .$
$x = - 3, y = - 5 or x = 3, y = 5$
$Therefore, the fixed points are (- 3, - 5) or (3, 5)$

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