Question

If $4y^2-32x+4y+65=0$ is the equation of a parabola, then which of the following is true?

• A. Its vertex is $(2,-\frac{1}{2})$
• B. Its axis is $x-2=0$
• C. Its focus is $(10,-\frac{1}{2})$
• D. Its axis is $2y+1=0$

Answer 1

Answer: (A), (D)

Its vertex is $(2,-\frac{1}{2})$
Its axis is $2y+1=0$

$4y^2-32x+4y+65=0$
$\Rightarrow (2y+1{)}^2=32(x-2)\Rightarrow {(y+\frac{1}{2})}^2=8(x-2)$
$\therefore$ Its vertex is given by $2y+1=0,x-2=0$
i.e.,$(2,-\frac{1}{2})$
Axis is $2y+1=0.$ Focus lies at a distance $a$ from the vertex on the axis $2y+1=0$
$\therefore$ Its coordinates are,
$(2+\frac{1}{4}\times 8,-\frac{1}{2})=(4,-\frac{1}{2})$