Question

A parabola whose focus is $S (3,4)$ is touching the coordinate axes. Then which of the following statement(s) is(are) correct.

• A. The equation of circle whose diameter is the portion of the tangent at the vertex of the parabola between the coordiante axes is $x^{2}+y^{2}-6 x-8 y=0$.
• B. The equation of circle whose diameter is the portion of the tangent at the vertex of the parabola between the coordinate axes is $x^{2}+y^{2}-3 x-4 y=0$.
• C. If $P$ and $Q$ are ends of a focal chord of the parabola then $\frac{1}{SP}+\frac{1}{SQ}$ equals $\frac{12}{5}$.
• D. If $P$ and $Q$ are ends of a focal chord of the parabola, then $\frac{1}{S P}+\frac{1}{S Q}$ equals $\frac{5}{12}$.

Answer 1

Answer: (B), (D)

Circle circumsribed the three tangents to the parabola always passes through the focus
$\therefore $ the circle passes through $S (3,4)$
Also perpendicular dropped from focus to any tangent meet the tangent at the vertex Figure AOBS is a rectangle then $A$ is $(3,0) \& B$ is $(0,4)$
$\therefore $ Equation of circle with $AB$ as diameter $x ^{2}+ y ^{2}-3 x -4 y =0 \Rightarrow B$

For(D)
$\frac{1}{SP}+\frac{1}{SQ}=\frac{1}{a} $ where $a$ is the distance of focus from tangent at the vertex
(semi latus rectum is the harmonic mean between the two focal segments.