Find the locus of the point of intersection of lines $x cos a + y sin a = a$ and $x sin a - y cos a = b$ ( $a$ is a variable).

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Solution

Let P(x,y) be the point of intersection given lines
$x cos a + y sin a = a . . . . . . . (1)$
$x sin a + y cos a = b . . . . . . . (2)$
Here is a variable , to eliminate it squaring and adding both side
$(x cos a + y sin a)^{2} + (x sin a - y sin a)^{2} = a^{2} + b^{2}$
$x^{2} {cos}^{2} a + y^{2} {sin}^{2} + 2 a b sin a cos a + x^{2} {sin}^{2} a + y^{2} {cos}^{2} a - 2 a b sin a cos a = a^{2} + b^{2}$
$x^{2} ({cos}^{2} a + {sin}^{2}) + y^{2} ({sin}^{2} a + {cos}^{2} a) = a^{2} + b^{2}$
$x^{2} + y^{2} = a^{2} + b^{2}$
Hence locus of $(x, y)$ is $x^{2} + y^{2} = a^{2} + b^{2}$

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