If the length of the latus rectum of a parabola, whose focus is (a,a) and the tangent at i…

Question

If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is $16$, then $|a|$ is equal to:

Accepted Answer

Equation of tangent at vertex : $$L \equiv x + y - a = 0$$

Focus : $$F \equiv (a,a)$$

Perpendicular distance of L from F

$$ = \left| {{{a + a - a} \over {\sqrt 2 }}} \right| = \left| {{a \over {\sqrt 2 }}} \right|$$

Length of latus rectum $$ = 4\left| {{a \over {\sqrt 2 }}} \right|$$

Given $$4\,.\,\left| {{a \over {\sqrt 2 }}} \right| = 16$$

$$ \Rightarrow |a| = 4\sqrt 2 $$

Solutions · 1