Question

Assertion :A chord of the curve $3x^2-y^2-2x+4y=0$ passing through the point $(1,-2)$ subtend a right angle at the origin. Reason: Lines represented by the equation $(3c+2m)x^2-2(1+2m)xy+(4-c)y^2=0$ are perpendicular if $c+m+2=0.$

• 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

Reason is true as the sum of the coefficients of $x^2$ and $y^2=3c+2m+4-c=0$

$\Rightarrow c+m+2=0$, so the lines are perpendicular if $c+m+2=0$
In assertion, let the equation of the chord by $y=mx+c$

Then equation of the pair of lines joining the origin to the points of intersection of the chord and the curve is
$3x^2-y^2-2x\left(\frac{y-mx}{c}\right)+4y\left(\frac{y-mx}{c}\right)=0$
$\Rightarrow (3c+2m)x^2-2(1+2m)xy+(4-c)y^2=0$
which are at right angles, if $c+m+2=0$. (using reason)

And since the line $y=mx+c$ passes through $(1,-2)$ $c+m+2=0$.

So, assertion is also true.