Prove that each of the lines given by the equation $(x_{1} y - x y_{1})^{2} = a^{2} (x^{2} + y^{2})$ is at a distance $a$ from the point $(x_{1}, y_{1})$ .

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Solution

Suppose that any of the straight lines represented by
homogeneous equation is $y - m x = 0$ .
$∴ m = y / x$ .
By the given condition $\frac{y_{1} - m x_{1}}{\sqrt (1 + m^{2})} = a$
or $(y_{1} - m x_{1})^{2} = a^{2} (1 + m^{2})$
Since $(2)$ is quadratic in $m$ , hence there will be
two straight lines which will satisfy the given condition.
In order to find their equation we will eliminate $m$
between $(1)$ and $(2)$ .
$∴ (y_{1} - \frac{y}{x} . x_{1})^{2} = a^{2} (1 + \frac{y^{2}}{x^{2}})$
or $(x y_{1} - y x_{1})^{2} = a^{2} (x^{2} + y^{2})$ .

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