Question

Assertion :Any chord of ellipse $x^2+y^2+xy=1$ through origin is bisected at origin. Reason: The centre of the ellipse is a point through which every chord is bisected.

• 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

Any chord passing through origin is $y=mx.$
So solving $y=mx$ & $x^2+y^2+xy=1$ we have $x^2+m^2{x}^2+x\left(mx\right)-1=0$ $x^2(1+m+m^2)-1=0$ Let $x_1,x_2$ be its roots. $\therefore x_1+x_2=0i.e.\frac{x_1+x_2}{2}=0$ and again $\frac{y_1+y_2}{2}=\frac{m{x}_1+m{x}_2}{2}$
$=m\left(\frac{x_1+x_2}{2}\right)=0$

$\Rightarrow$Mid point of chord is origin for all values of $m.$