Question

Assertion :$PQR$ is a right angled isosceles triangle, right angled at $P(2,1)$. Equation of the line $QR$ is $2x+y=3$.

Equation representing the pair of lines $PQ$ and $PR$ is $3{(x-2)}^2-8(x-2)(y-1)-3{(y-1)}^2=0$

Reason: If the origin is shifted to the point $P(2,1)$, then equation representing $PQ$ and $PR$ in the new coordinate system is $xy=0$

• 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 C Assertion is correct but Reason is incorrect

$PQ,PR$ subtend equal angle $\pi /4$ with the hypotenuse $QR$. Let the slope of $PQ$ be m. then as the slope of $QR$ is $-2$
$\frac{m+2}{1-2m}=\pm 1\Rightarrow 3m^2-8m-3=0$
$\Rightarrow 3{(y-1)}^2-8(y-1)(x-2)-3{(x-2)}^2=0$ as the equation of $PQ$ is $y-1=m(x-2)$
which shows that assertion is true .
If the origin is shifted to the point $P(2,1)$. Equation of the lines $PQ$ and $PR$ in the new system is
$3y^2-8xy-3x^2=0$ so the reason is false