Question

Equation $x^2-2x-2y+5=0$ represents

• A. a parabola with vertex $(1,2)$
• B. a parabola with vertex $(2,1)$
• C. a parabola with directrix $y=\frac{3}{2}$
• D. a parabola with directrix $y=\frac{2}{5}$

Answer 1

Answer: (A), (C)

a parabola with vertex $(1,2)$
a parabola with directrix $y=\frac{3}{2}$

Given conic is, $x^2-2x-2y+5=0$
$x^2-2x+1=2y-4$
$(x-1{)}^2=2(y-2)$
Shifting origin to $(1,2)$
$X^2=2Y$ where $X=x-1$ and $Y=y-2$
Clearly this is Parabola. with latus rectum $2$ (here $a=\frac{1}{2})$
Thus vertex of the parabola is $X=0,Y=0\Rightarrow (x=1,y=2)$
And directrix of $X^2=4aY$ is $Y=-a\Rightarrow y-2=-\frac{1}{2}\Rightarrow y=\frac{3}{2}$