First slide

Equation of line in Straight lines

Question

$P$ and $Q$ are two variable points on the axes of $x$ and $y$ respectively such that $| O P | + | O Q | = α$ , then the locus of foot of perpendicular from $(0, 0)$ on $P Q$ is

Moderate

A

$(x + y) (x^{2} + y^{2}) = a (x - y)$
B

$(x - y) (x^{2} - y^{2}) = a x y$
C

$(x + y) (x^{2} + y^{2}) = a x y$
D

$(x + y) (x^{2} - y^{2}) = a x y$

Solution

Let $\bar{P Q}$ be $\frac{x}{α} + \frac{y}{β} = 1$ equation of the circle passing through $O, P, R$ is
$x^{2} + y^{2} - α x = 0 . \Rightarrow α = \frac{x^{2} + y^{2}}{x}$ where Ris foot of perpendicular
Similarly $β = \frac{x^{2} + y^{2}}{y}$ $∴$ $| α | + | β | = a$
Now eliminate $α, β$

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