Question

The equation $y^2-8y-x+19=0$ represents

• A. a parabola whose focus is $\left(\frac{1}{4},0\right)$ and directrix is $x=-\frac{1}{4}$
• B. a parabola whose vertex is (3, 4) and directrix is $x=\frac{11}{4}$
• C. a parabola whose focus is $\left(\frac{13}{4},4\right)$ and vertex is (0, 0)
• D. a curve which is not a parabola

Answer 1

Answer: (B)

Given equation is $y^2-8y-x+19=0$
$\Rightarrow$ ${(y-4)}^2=x-19+16$
$\Rightarrow$ ${(y-4)}^2=(x-3)$
$\Rightarrow$ $y^2=4AX$
where $Y=y-4,A=\frac{1}{4}$
and $X=x-3$ $\therefore$
Focus $=(A,0)=\left(\frac{1}{4},0\right)=\left(\frac{13}{4},4\right)$
Vertex = (3, 4) Directrix, $X=-\frac{1}{4}$
$\Rightarrow$ $x-3=-\frac{1}{4}$
$\Rightarrow$ $x=\frac{11}{4}$