# Find the equation of a parabola, whose vertex is at (-3,2) and passes through (4,7)?

Apr 5, 2017

$5 {x}^{2} + 30 x - 49 y + 143 = 0$ or $7 {y}^{2} - 25 x - 28 y - 47 = 0$

#### Explanation:

There could be two type of parabolas with vertex as $\left(- 3 , 2\right)$

A $\left(y - 2\right) = a {\left(x + 3\right)}^{2}$ If this passes through $\left(4 , 7\right)$, we have

$\left(7 - 2\right) = a {\left(4 + 3\right)}^{2}$ i.e. $a = \frac{5}{49}$ and equation of parabola is

$\left(y - 2\right) = \frac{5}{49} {\left(x + 3\right)}^{2}$ i.e. $49 y - 98 = 5 {x}^{2} + 30 x + 45$ or

$5 {x}^{2} + 30 x - 49 y + 143 = 0$

B $\left(x + 3\right) = a {\left(y - 2\right)}^{2}$ If this passes through $\left(4 , 7\right)$, we have

$\left(4 + 3\right) = a {\left(7 - 2\right)}^{2}$ i.e. $a = \frac{7}{25}$ and equation of parabola is

$\left(x + 3\right) = \frac{7}{25} {\left(y - 2\right)}^{2}$ i.e. $25 x + 75 = 7 {y}^{2} - 28 y + 28$ or

$7 {y}^{2} - 25 x - 28 y - 47 = 0$

graph{(5x^2+30x-49y+143)(7y^2-25x-28y-47)=0 [-11, 9, -1.36, 8.64]}