Question

Assertion :If $PSQ$ is the focal chord of the parabola $y^2=8x$ such that $SP=6$, then the length $SQ$ is $3$. Reason: The semi-latus rectum of a parabola is the harmonic mean between the segment of any focal chord of a parabola.

• 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. Assertion is incorrect but Reason is correct

Answer 1

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

Let $P=(a{t}^2,2at),\therefore Q^{\prime }=(\frac{a}{t^2},-\frac{2a}{t})$
$S=(a,0),L=(a,2a)$
Now $SP=\sqrt{(a-a{t}^2{)}^2+(2at{)}^2}=a+a{t}^2$
$SQ=\sqrt{{(a-\frac{a}{t^2})}^2+{\left(\frac{-2a}{t}\right)}^2}=a+\frac{a}{t^2}=\frac{a+a{t}^2}{t^2}$
Now clearly,

$\frac{1}{SP}+\frac{1}{SQ}=\frac{2}{SL}$
Hence, $SP,SL,SQ$ are in H.P.
Assertion: $SP=3,SL=2a=4$
$\Rightarrow \frac{1}{6}+\frac{1}{SQ}=\frac{1}{2}\Rightarrow SQ=3$
Hence both statements are true and Reason is correct explanation of Assertion.