Question

Find the vertex, axis, focus, directrix and latus rectum of the parabola, also draw their rough sketches
$4y^2+12x-20y+67=0$

Answer 1

The given equation is
$4y^2+12x-20y+67=0$ $\Rightarrow y^2+30x-5y+\frac{67}{4}=0$
$\Rightarrow y^2-5y=-3x-\frac{67}{4}$ $\Rightarrow y^2-5y+{\left(\frac{5}{2}\right)}^2=-3x-\frac{67}{4}+{\left(\frac{5}{2}\right)}^2$
$\Rightarrow {(y-\frac{5}{2})}^2=-3x-\frac{42}{4}$ $\Rightarrow {(y-\frac{5}{2})}^2=-3(x+\frac{7}{2})...........\left(i\right)$
Let $x=X-\frac{7}{2},y=Y+\frac{5}{2}............\left(ii\right)$
Using these relations equation (i) reduces to $y^2=-3X...........\left(iii\right)$
This is of the form $y^2=-4aX$ On comp0aring We get 4a = 3 $\Rightarrow$ a = 3/4
Vertex - The coordinates of the vertex are (X = 0, Y = 0)
So the coordinates of the vertex are $(-\frac{7}{2},\frac{5}{2})$ $[PuttingX=0,Y=0in(ii\left)\right]$