Question

What is the equation of the latus rectum of the parabola $x^2+4x+2y=0$?

• A. $2y+3=0$
• B. $3y=2$
• C. $2y=3$
• D. $3y+2=0$

Answer 1

The correct option is C

$2y=3$

Explanation for correct option:

Finding the equation of latus rectum:

Given, equation of parabola is $x^2+4x+2y=0$

Simplifying the given equation, we get

$\Rightarrow x^2+4x=-2y$

Adding $4$ both sides, to make L.H.S term a perfect square

$$\begin{gathered} \Rightarrow x^2+4x+4=-2y+4 \\ \Rightarrow {\left(x+2\right)}^2=-2(y-2)\left[\because {(x+2)}^2=x^2+4x+4\right] \\ \Rightarrow {\left(x+2\right)}^2=-4\left(\frac{1}{2}\right)(y-2)\dots \dots \left(1\right) \end{gathered}$$

This equation is of the form of parabola $X^2=-4aY\dots \dots \left(2\right)$

we know that,

For the downward parabolas, the equation of latus rectum is $Y=-a$

Comparing the equations $\left(1\right)\text{and}\left(2\right)$, we get

$X=x+2,Y=y-2,a=\left(\frac{1}{2}\right)$

Then , the equation of latus rectum,

$\begin{array}{rcl}Y& =& -a\\ \left(y-2\right)& =& -\frac{1}{2}\\ y& =& \frac{3}{2}\end{array}$

Thus, the equation of latus rectum is $2y=3$.

Hence, option(C) is the correct answer.