Question

Find the vertex, axis, latus rectum, and focus of the parabolas $x^2+2y=8x-7.$

Answer 1

$x^2+2y=8x-7x^2-8x=-2y-7x^2-8x+16=-2y-7+16{(x-4)}^2=-2(y-\frac{9}{2})$

Comparing with the standard equation $x^2=4ay$

Vertex of the parabola is

$x-4=0,y-\frac{9}{2}=0\Rightarrow x=4,y=\frac{9}{2}(4,\frac{9}{2})$

Axis of the parabola is

$x-4=0$

$x=4$

Latus rectum of the parabola is $|4a|$

Here $4a=-2$

$\Rightarrow a=-\frac{1}{2}\Rightarrow L{L}^{\prime }=2$

Focus of the parabola is

$x-4=0,y-\frac{9}{2}=-\frac{1}{2}\Rightarrow x=4,y=4(4,4)$