Question

Find the vertex, axis, latus rectum, and focus of the parabolas $x^2-2ax+2ay=0.$

Answer 1

$x^2-2ax+2ay=0x^2-2ax=-2ay{x}^2-2ax+a^2=-2ay+a^2{(x-a)}^2=-2a(y-\frac{a}{2})$

Comparing with the standard equation $X^2=4AY$

Vertex of the parabola is $X=0,Y=0$

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

Axis of the parabola is $X=0$

$x-a=0$

$x=a$

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

Here $A=-\frac{a}{2}$

$\Rightarrow L{L}^{\prime }=2a$

Focus of the parabola is $X=0,Y=A$

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