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} $