Question

Find the equation of the parabola passing through the points

(3, 0), (- 3, 0)

and

(2, 5)

and having its axis parallel to the

y

-axis. Also find the co-ordinates of its vertex.

Accepted Answer

Given the axis of parabola is parallel to $y$ -axis.

We know that the equation of parabola which has axis parallel to $y$ -axis say $x = h$ is given by

${(x - h)}^{2} = 4 a (y - k) . . . . (1)$

Where $(h, k)$ is the vertex and $4 a$ is length of latus rectum.

Also, given the parabola passes through $(3, 0), (- 3, 0) & (2, 5)$

Putting $(3, 0), (- 3, 0) & (2, 5) in (1)$

We get ${(3 - h)}^{2} = 4 a (0 - k)$

$\Rightarrow {(3 - h)}^{2} = - 4 a . . . . (2)$

${(- 3 - h)}^{2} = 4 a (0 - k)$

$\Rightarrow {(- 3 - h)}^{2} = - 4 a k . . . . (3)$

And $y$

Dividing $(2) by (3)$

$\frac{{(3 - h)}^{2}}{{(- 3 - h)}^{2}} = \frac{- 4 a k}{- 4 a k}$

$\Rightarrow {(3 - h)}^{2} = {(- 3 - h)}^{2}$

$\Rightarrow 9 + h^{2} - 6 h = 9 + h^{2} + 6 h$

$\Rightarrow 12 h = 0$

$\Rightarrow h = 0$

Now dividing $(2) by (4)$ we gate

$\frac{{(3 - h)}^{2}}{{(2 - h)}^{2}} = \frac{- 4 a k}{4 a (5 - k)}$

$\Rightarrow \frac{{(3 - 0)}^{2}}{{(2 - 0)}^{2}} = \frac{- k}{5 - k}$

$\Rightarrow \frac{9}{4} = \frac{- k}{5 - k}$

$\Rightarrow 9 (5 - k) = - 4 k$

$\Rightarrow 45 - 9 k = - 4 k$

$\Rightarrow 45 = 9 k - 4 k$

$\Rightarrow 45 = 5 k \Rightarrow k = 9$

Putting the value of $h & k$ in $(2)$

${(3 - 0)}^{2} = - 4 a \cdot 9$

$\Rightarrow 9 = - 4 a \cdot 9$

$\Rightarrow a = \frac{- 1}{4}$

Now put the values of $h, k$ and da in $(1)$

We get

${(x = 0)}^{2} = 4 (\frac{- 1}{4}) (y - 9)$

$\Rightarrow x^{2} = - (y - 9)$

$\Rightarrow x^{2} = - y + 9$

And $x^{2} + y = 9$

Which is the equation of required parabola.

And thee coordinates of vertex are $(0, 9)$

Find the equation of the parabola passing through the points $(3, 0), (- 3, 0)$ and $(2, 5)$ and having its axis parallel to the $y$ -axis. Also find the co-ordinates of its vertex.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

Find the equation of the parabola passing through the points 3, 0, -3, 0 and 2, 5 and having its axis parallel to the y-axis. Also find the co-ordinates of its vertex.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

Find the equation of the parabola passing through the points $(3, 0), (- 3, 0)$ and $(2, 5)$ and having its axis parallel to the $y$ -axis. Also find the co-ordinates of its vertex.