Question

Assertion :Length of latus rectum of the parabola ${(2x+3y+2)}^2=4(3x+2y+3)$ is $4$. Reason: Length of latus rectum of the parabola $y^2=4ax$ is $4a$.

• 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 and Reason is correct

Answer 1

The correct option is D Assertion is incorrect and Reason is correct

$(2x+3y+2{)}^2=4(3x+2y+3)$
Let$,Y=(2x+3y+2)\to \left(1\right)$ and $X=(3x+2y+3)\to \left(2\right)$
The given parabola becomes $Y^2=4X$
So, the coordinates of the end points of the latus rectum are
$(X=1,Y=2)$ and $(X=1,Y=-2)$
So, solving $e{q}^{n}s\left(1\right)$ and $\left(2\right)$ by putting corresponding values of X and Y, we get
$(x=-\frac{6}{5},y=\frac{4}{5})$ and $(x=\frac{2}{5},-\frac{8}{5})$
The length of the latus rectum is$\frac{\sqrt{(2+6{)}^2+(4+8{)}^2}}{5}=\frac{\sqrt{208}}{5}\ne 4$
So, the only option that can be correct is (D).