Mathematics - Parabola Question with Solution | TestHub

$\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$ For this parabola x is a tangent at $\mathrm{P}(\mathrm{a},0)$ Y-axis a tangent $\mathrm{Q}(0,\mathrm{b})$ $\therefore \mathrm{O}(0,0)$ is point if inter section perpendicular tagents ∴ directrix passing through this point Clearly OSP $=90^0$ Hence circle on OP as diameter passing though S i.e., $x^2+y^2-ax=0$ passing through S. Ily, $OSQ=90^0\therefore x^2+y^2-bx=0$ passing through S . Point of intersecting above circle is focus. $\begin{array}{rlr}& x^2+y^2-ax=0& x^2+y^2-bx=0\end{array}$ $ax-by=0$ $\begin{array}{c}y=\frac{ax}{b}\Rightarrow x^2+\frac{a^2{x}^2}{b^2}=axx\left(\frac{b^2+a^2}{b^2}\right)=a\end{array}$

$x=\frac{a{b}^2}{a^2+b^2}$ Ily, $y=\frac{a^{2}b}{a^2+b^2}$ focus $S=\left(\frac{a{b}^2}{a^2+b^2},\frac{a^{2}b}{a^2+b^2}\right)$.