Question

Assertion :The conic $\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$ represents a parabola, where $a,b\ne 0$. Reason: The second degree equation $A{x}^2+2Hxy+B{y}^2+2Gx+2Fy+c=0$ represents a parabola if $\Delta \ne 0$ & $H^2=AB$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$

On squaring both sides we get $\frac{x}{a}+\frac{y}{b}+2\sqrt{\frac{xy}{ab}}=1$ $\therefore \frac{x}{a}+\frac{y}{b}-1=-2\sqrt{\frac{xy}{ab}}$

Again squaring both sides, we get $\frac{x^2}{a^2}+\frac{y^2}{b^2}+1+\frac{2xy}{ab}-\frac{2x}{a}-\frac{2y}{b}=\frac{4xy}{ab}$
$\Rightarrow \frac{x^2}{a^2}-\frac{2xy}{ab}+\frac{y^2}{b^2}-\frac{2x}{a}-\frac{2y}{b}+1=0$

Now comparing with the equation given in Reason (R) $\therefore A=\frac{1}{a^2},B=\frac{1}{b^2},H=-\frac{1}{ab},G=-\frac{1}{a},F=-\frac{1}{b},C=1$
$\Delta =ABC+2FGH-A{F}^2-B{G}^2-C{H}^2$
$=\frac{1}{a^2{b}^2}-\frac{2}{a^2{b}^2}-\frac{1}{a^2{b}^2}-\frac{1}{a^2{b}^2}-\frac{1}{a^2{b}^2}=\frac{-4}{a^2{b}^2}\ne 0$ as $a\ne 0,b\ne 0$ & $H^2=AB$

$\because \Delta \ne 0,H^2=AB$ $\therefore$Conic is a parabola