Question

Find standard form of the equation of the parabola that satisfies the given conditions:Directrix: x = -4Focus: (2, 4)

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given data

`\begin{gathered} \text{directrix}=x=-4 \\ \text{Focus:}(2,4) \end{gathered}`

STEP 2: Write the equation of a parabola

`\begin{gathered} \text{The equation is given as:} \\ x=(1)/(4(f-h))(y-k)^2+h\text{ where} \\ (h,k)\text{ is the vertex} \\ (f,k)is\text{ the focus} \\ \text{Thus,} \\ f=2,k=4 \end{gathered}`

STEP 3: Get the value of h

The distance from the focus to the vertex is equal to the distance from the vertex to the directrix. Therefore:

`\begin{gathered} f-h=h-(-4) \\ By\text{ substitution}, \\ 2-h=h+4 \\ 2-4=h+h \\ -2=2h \\ h=-(2)/(2)=-1 \end{gathered}`

STEP 4: Get the standard form of equation

Hence, the standard form becomes:

`\begin{gathered} \text{The standard form is given as:} \\ x=(y^2)/(12)-(2y)/(3)+(1)/(3) \end{gathered}`