First slide

Equation of line in Straight lines

Question

If the quadrilateral formed by the lines $ax + by + c = 0$ , $a' x + b' y + c = 0,$ $ax + by + c' = 0$ , $a' x' + b' y + c' = 0$ has perpendicular diagonals, then

Moderate

A

$b^{2} + c^{2} = b'^{2} + c'^{2}$
B

$c^{2} + a^{2} = c'^{2} + a'^{2}$
C

$a^{2} + b^{2} = a'^{2} + b'^{2}$
D

none of these

Solution

Since the diagonals are perpendicular, the given quadrilateral is a rhombus. So, the distances between two pairs of parallel sides are equal. Hence,
$|\frac{c' - c}{\sqrt{a^{2} + b^{2}}}| = |\frac{c' - c}{\sqrt{a'^{2} + b'^{2}}}|$
or $a^{2} + b^{2} = a'^{2} + b'^{2}$

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