Question

Write the equation of the parabola with vertex (3, 4) and focus (5, 4).

Answer 1

Answer:

The equation of the parabola with vertex (3, 4) and focus (5, 4) will be:

• $\left(4\:-\:k\right)^2\:=\:8\left(3\:-\:h\right)\:$

Step-by-step explanation:

Given

• vertex (3, 4)

• focus (5, 4)

As the y value of the vertex (3, 4) and the focus (5, 4) is the same.

So the parabola opens to the right.

Thus

The standard form of Parabola

$\left(y\:-\:k\right)^2\:=\:4p\left(x\:-\:h\right)\:$

Substituting vertex

$\left(y\:-\:4\right)^2\:=\:4p\left(x\:-\:3\right)\:$

As focus = (h + p, k)

5 = 3 + p

p = 2

so

$\left(y\:-\:4\right)^2\:=\:4p\left(x\:-\:3\right)\:$

$\left(y\:-\:4\right)^2\:=\:4\left(2\right)\left(x\:-\:3\right)\:$

$\left(y\:-\:4\right)^2\:=\:8\left(x\:-\:3\right)\:$

Therefore, the equation of the parabola with vertex (3, 4) and focus (5, 4) will be:

• $\left(y\:-\:4\right)^2\:=\:8\left(x\:-\:3\right)\:$

Answer 2

The equation of the parabola with vertex (3, 4) and focus (5, 4) is $(4-k)^2 - 8(3 - h)$.

Calculation of the equation:

Since

The vertex (3,4) and there is the focus (5,4)

Here the parabola should be opened to the right.

So,

the standard form should be

$(y -k)^2 = 4p(x - h)\\\\(y - 4)^2 = 4p(x - 3)$

Since

focus = (h + p, k)

5 = 3 + p

p = 2

So,

$(y - 4)^2 = 4p(x - 3)\\\\(y - 4)^2 = 4(2) (x - 3)\\\\(y - 4)^2 = 8(x - 3)$

Hence, The equation of the parabola with vertex (3, 4) and focus (5, 4) is $(4-k)^2 - 8(3 - h)$.

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