Question

Find the equation of the parabola with a focus (2, 5) and directrix of y = 3.

Answer 1

Answer:

One form of an equation is given by  

y=a(x−h

)

2

+k

y=a(x−h)2+k

where  

(h,k)

(h,k)

the coordinates of the vertex and  

(h,k+

1

4a

)

The parabola is symmetric with respect to  

y=x

y=x

and can be viewed as a standard downward parabola with a rotation of 45 degrees clockwise. So, follow the steps below to obtain its equation.

1) The length between the focus and vertex is  

f=

4

2

+

4

2

−

−

−

−

−

−

√

=4

2

–

√

f=42+42=42

. The standard equation is  

y=−

1

4f

x

2

y=−14fx2

.

2) Shift the vertex to (-2, -2),

y+2=−

1

16

2

–

√

(x+2

)

2

y+2=−1162(x+2)2

3) Rotate the equation -45-degrees  

x→

1

2

√

(x−y)

x→12(x−y)

,  

y→−

1

2

√

(x+y)

y→−12(x+y)

to get

−

1

2

–

√

(x+y)+2=−

1

16

2

–

√

(

1

2

–

√

(x−y)+2)

2

Step-by-step explanation: