Question

For the parabola
$x^2+8x+12y+4=0$, which of the following options is/are correct?

• A. length of latus rectum is $12$ units
• B. coordinates of vertex are $(-4,1)$
• C. length of latus rectum is $16$ units
• D. coordinates of vertex are $(4,2)$

Answer 1

Answer: (A), (B)

coordinates of vertex are $(-4,1)$

The equation of parabola is
$x^2+8x+12y+4=0\Rightarrow (x+4{)}^2-16+12y+4=0\Rightarrow (x+4{)}^2-12+12y=0(x+4{)}^2=-12y+12\Rightarrow (x+4{)}^2=-12(y-1)$
Let $x+4=X,y-1=Y$
$\therefore X^2=-12Y$
Comparing it with $x^2=-4ay$, we get
$\therefore a=3$
and vertex as $(X,Y)\equiv (0,0)$
$\Rightarrow X=0,Y=0$
$\Rightarrow x+4=0,y-1=0$
$\therefore x=-4,y=1$

Vertex of given parabola is $(-4,1)$

Length of latus rectum$=4a=4\times 3=12$