#### Solved Examples and Worksheet for Standard Forms and Equations of Ellipses and Parabolas

Q1Find an equation of the ellipse with foci (0, - 4) and (0, 4) whose minor axis has a length of 8 units.
A. x232-y216 = 1
B. 32x2 + 16y2 = 1
C. x232+y216 = 1
D. y232+x216 = 1

Step: 1
The foci of the ellipse are (0, - 4) and (0, 4), which are on the y - axis with c = 4.
Step: 2
The center of the ellipse is (0 + 02, - 4 + 42) = (0, 0).
[Midpoint of the line segment joining foci.]
Step: 3
Length of the semiminor axis is b = 82 = 4
Step: 4
a2 = b2 + c2 = 16 + 16 = 32
[Pythagorean relation.]
Step: 5
So, the standard equation of the ellipse is y232+x216 = 1
Correct Answer is :    y232+x216 = 1
Q2Identify an equation of the parabola in the standard form whose vertex is (0, 0) and directrix is y = - 7.

A. y2 = 28x
B. x2 = - 28y
C. x2 = 7 y
D. x2 = 28y

Step: 1
Vertex of the parabola = (0, 0)
Step: 2
Equation of the directrix is y = - 7
Step: 3
Since, y = - 7 is perpendicular to x - axis, x - axis is the focal axis of the parabola.
Step: 4
p = 7
[Find p by comparing y = - 7 with y = - p.]
Step: 5
So, the equation of the parabola in the standard form is x2 = 4py or x2 = 4(7)y x2 = 28y.
[Substitute p = 7.]
Correct Answer is :   x2 = 28y
Q3Find an equation of the ellipse with foci (0, - 3) and (0, 3) whose minor axis has a length of 6 units.
A. 18x2 + 9y2 = 18
B. x29+y218 = 1
C. 9x2 + 18y2 = 1
D. x218+y29 = 1

Step: 1
The foci of the ellipse are (0, - 3) and (0, 3) which are on the y - axis with c = 3.
Step: 2
The center of the ellipse = (0 + 02, -3 + 32) = (0, 0).
[Use the mid point formula.]
Step: 3
Length of the semi minor axis is b = 62 = 3
Step: 4
a2 = b2 + c2 = 9 + 9 = 18
[Pythagorean relation.]
Step: 5
So, the standard equation of the ellipse is x29+y218 = 1.
Correct Answer is :    x29+y218 = 1
Q4Find an equation in the standard form for the ellipse with foci (6, - 7), (8, - 7) and the end points of whose major axis are (5, - 7) and (9, - 7).
A. (x - 7)23+(y + 7)24 = 1
B. (x - 7)24+(y + 7)23 = 1
C. 7x2 + y2 = 1
D. (x - 7)23+(y - 7)24 = 1

Step: 1
Foci of the ellipse are (6, - 7) and (8, - 7).
Step: 2
Since the y - coordinates of foci are same, the focal axis of the ellipse is parallel to x - axis.
Step: 3
Distance between the foci = 2c = (8 - 6)2+(- 7 + 7)2 = 2 and hence c = 1
Step: 4
End points of the major axis are (5, - 7) and (9, - 7).
Step: 5
Distance between (5, - 7) and (9, - 7) = 2a = (9 - 5)2 +(- 7 + 7)2 = 4
Step: 6
So, length of the semi major axis = a = 42 = 2
Step: 7
b2 = a2 - c2 = 4 - 1 = 3
[Pythagorean relation.]
Step: 8
Center of the ellipse, (h, k) = (5 + 92, - 7 - 72 ) = (7, - 7)
[Center of the ellipse is the mid point of its major axis.]
Step: 9
So, the equation of the ellipse is (x - 7)24+(y + 7)23 = 1.
[Use the standard form of the ellipse is (x - h)2a2+(y - k)2b2 = 1.]
Correct Answer is :   (x - 7)24+(y + 7)23 = 1
Q5What is the standard form of the equation of a parabola with focus (6, 0), directrix x = - 6?
A. y = 24x2
B. y2 = 4x
C. y2 = 24x
D. y2 = - 24x

Step: 1
The equation of a parabola with focus (a, 0) and directrix x = - a is y2 = 4ax.
Step: 2
y2 = 4(6) x
[Replace a with 6.]
Step: 3
y2 = 24x
Correct Answer is :    y2 = 24x
Q6Given the parabola with focus (0, 6) and directrix y = - 6. Determine its equation and which direction it opens.

A. x2 = 6y, downwards
B. x2 = 6y, upwards
C. x2 = 24y, upwards
D. x2 = 24y, downwards

Step: 1
A parabola with focus (0, c) and directrix y = - c, equation is given by x2 = 4cy.
Step: 2
Since c = 6 > 0, it open upwards.
Step: 3
x2 = 4(6)y = 24y
[Replace c with 6.]
Step: 4
So, the equation of parabola is x2 = 24y and it open upwards.
Correct Answer is :   x2 = 24y, upwards
Q7Find an equation of the parabola in the standard form whose directrix is the line y = 9 and whose focus is the point (0, - 9).
A. y2 = - 36x
B. x2 = 36y
C. y2 = 36x
D. x2 = - 36y

Step: 1
The equation of a parabola in the standard form whose directrix is the line y = p and focus is the point (0, - p) is x2 = - 4py.
Step: 2
The directrix is y = 9 and the focus is (0, - 9). So, p = 9.
Step: 3
The equation of the parabola in the standard form is x2 = - 36y.
[Replace p with 9.]
Correct Answer is :   x2 = - 36y
Q8Given the parabola with focus (0, 3) and directrix y = - 3. Determine its equation and which direction it opens.
A. x2 = 3y, upwards
B. x2 = 12y, downwards
C. x2 = 3y, downwards
D. x2 = 12y, upwards

Step: 1
A parabola with focus (0, c) and directrix y = - c, equation is given by x2 = 4cy.
Step: 2
Since c = 3 > 0, it open upwards.
Step: 3
x2 = 4(3)y = 12y
[Replace c with 3]
Step: 4
So, the equation of parabola is x2 = 12y and it open upwards.
Correct Answer is :   x2 = 12y, upwards
Q9Find an equation in the standard form for the ellipse with foci (4, - 8), (10, - 8) and the end points of whose major axis are (5, - 8) and (15, - 8).
A. (x - 10)25+(y - 8)23 = 1
B. (x - 10)216+(y + 3)225 = 1
C. (x - 10)225+(y + 8)216 = 1
D. 16x2 + y2 = 1

Step: 1
Foci of the ellipse are (4, - 8) and (10, - 8).
Step: 2
Since the y - coordinates of foci are same, the focal axis of the ellipse is parallel to x - axis.
Step: 3
Distance between the foci = 2c = (10 - 4)2+(- 8 + 8)2 = 6 and hence c = 3.
Step: 4
End points of the major axis are (5, - 8) and (15, - 8).
Step: 5
Distance between (5, - 8) and (15, - 8) = 2a = (15 - 5)2+(- 8 + 8)2 = 10
Step: 6
So, the length of the semimajor axis is a = 102 = 5
Step: 7
b2 = a2 - c2 = 25 - 9 = 16
[Pythagorean relation.]
Step: 8
Since the center of the ellipse is midpoint of its major axis, center of the ellipse is (h, k) = (5+152, - 8 - 82 ) = (10 , - 8).
Step: 9
So, the equation of the ellipse in the standard form is (x - h)2a2+(y - k)2b2 = 1 ⇒ (x - 10)225+(y + 8)216 = 1.
Correct Answer is :   (x - 10)225+(y + 8)216 = 1