Question

Find the equation of parabola with focus $(5,0)$ and vertex $(5,3)$.

• A. $y=12(x-5{)}^2-3$
• B. $y-3=-(x-5{)}^2$
• C. $y=\frac{(x-5{)}^2}{12}-3$
• D. $y=-\frac{(x-5{)}^2}{12}+3$

Answer 1

Answer: (D)

$y=-\frac{(x-5{)}^2}{12}+3$

Since the $x$-coordinates of the vertex and focus are the same, they are one of top of the other.

So, this is a regular vertical parabola, where the $x$ part is squared.

Since the vertex is below the focus, this is a right-side up parabola and '$p$' is positive.

Thus $(x-h{)}^2=4p(y-k)$

Focus $(5,0)$ vertex $(5,3)$

$\Rightarrow p=0-3$

$\Rightarrow p=-3$

Therefore, $(x-5{)}^2=4(-3\left)\right(y-3)$

$\Rightarrow -(x-5{)}^2/12=y-3$

$\Rightarrow -\left(\right(x-5{)}^2/12)+3=y$

Hence, option d is correct.