First slide

Equation of line in Straight lines

Question

If $p$ is the length of the perpendicular from the origin on the line $\frac{x}{a} + \frac{y}{b} = 1$ and $a^{2}, p^{2}, b^{2}$ are in A.P., then $a^{4} - 2 p^{2} a^{2} + 2 p^{4} =$

Moderate

A

-1
B

0
C

1
D

none of these

Solution

We have $p = \frac{| a b |}{\sqrt{a^{2} + b^{2}}} \Rightarrow p^{2} = \frac{a^{2} b^{2}}{a^{2} + b^{2}}$
Also, since $a^{2}, p^{2}, b^{2}$ are in A.P.,
$a^{2} + b^{2} = 2 p^{2}$ and $a^{2} b^{2} = p^{2} (a^{2} + b^{2}) = 2 p^{4}$
$\Rightarrow 2 p^{4} = a^{2} (2 p^{2} - a^{2}) \Rightarrow a^{4} - 2 p^{2} a^{2} + 2 p^{4} = 0$

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