Maths

Ellipse MCQ

October 1, 2023
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1. What is the equation of an ellipse with its major axis parallel to the x-axis?
a) \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
b) \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
c) x^2 - y^2 = a^2
d) y^2 - x^2 = a^2

2. The foci of an ellipse are:
a) Always located at the origin
b) Located at the endpoints of the major axis
c) Located at the endpoints of the minor axis
d) Located at the center of the ellipse

3. What is the eccentricity of a circle?
a) 0
b) 1
c) Between 0 and 1
d) Greater than 1

4. In the standard equation of an ellipse, which term is responsible for the horizontal stretch or compression?
a) a
b) b
c) c
d) \frac{1}{a}

5. If the major axis of an ellipse is vertical, what is the equation of the ellipse?
a) \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
b) \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
c) x^2 - y^2 = a^2
d) y^2 - x^2 = a^2

6. The length of the minor axis of an ellipse is:
a) Equal to 2a
b) Equal to 2b
c) Equal to a + b
d) Half of the length of the major axis

7. What is the center of the ellipse with the equation \frac{x^2}{4} + \frac{y^2}{9} = 1?
a) (0, 0)
b) (2, 3)
c) (4, 9)
d) (-2, -3)

8. If the eccentricity of an ellipse is 0, what kind of conic section is it?
a) Circle
b) Parabola
c) Hyperbola
d) Ellipse

9. The sum of the distances from any point on the ellipse to its foci is:
a) Equal to the length of the major axis
b) Equal to the length of the minor axis
c) Always constant
d) Equal to the eccentricity

10. The equation of the ellipse with foci at (±c, 0) and major axis along the x-axis is:
a) \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
b) \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
c) \frac{x^2}{c^2} + \frac{y^2}{b^2} = 1
d) \frac{x^2}{a^2} - \frac{y^2}{c^2} = 1

11. The equation of the ellipse with foci at (0, ±c) and major axis along the y-axis is:
a) \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
b) \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
c) \frac{x^2}{c^2} + \frac{y^2}{b^2} = 1
d) \frac{y^2}{a^2} - \frac{x^2}{c^2} = 1

12. The standard equation of an ellipse with center at (h, k) is:
a) \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
b) \frac{(x + h)^2}{a^2} + \frac{(y + k)^2}{b^2} = 1
c) \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1
d) \frac{(x + h)^2}{b^2} + \frac{(y + k)^2}{a^2} = 1

13. The length of the semi-major axis of an ellipse is denoted by:
a) a
b) b
c) c
d) d

14. The length of the semi-minor axis of an ellipse is denoted by:
a) a
b) b
c) c
d) d

15. If a > b in the equation of an ellipse, what is the shape of the ellipse?
a) Circle
b) Horizontal ellipse
c) Vertical ellipse
d) Parabola

16. The eccentricity of a circle is:
a) 0
b) 1
c) Between 0 and 1
d) Greater than 1

17. The value of c in the standard equation of an ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 is equal to:
a) a
b) b
c) \sqrt{a^2 - b^2}
d) \sqrt{b^2 - a^2}

18. The equation of the ellipse with foci at (±2, 0) and a = 3 is:
a) \frac{x^2}{3} + \frac{y^2}{4} = 1
b) \frac{x^2}{4} + \frac{y^2}{3} = 1
c) \frac{x^2}{9} + \frac{y^2}{4} = 1
d) \frac{x^2}{4} + \frac {y^2}{9} = 1

19. An ellipse becomes a circle when:
a) a = 0
b) a = b
c) a = 1
d) b = 1

20. The equation of an ellipse with a = 5 and b = 4 is:
a) \frac{x^2}{16} + \frac{y^2}{25} = 1
b) \frac{x^2}{25} + \frac{y^2}{16} = 1
c) \frac{x^2}{5} + \frac{y^2}{4} = 1
d) \frac{x^2}{4} + \frac{y^2}{5} = 1

Key

Protected: Ellipse MCQ Key

Key Concepts

  • An ellipse is a type of conic section, formed by slicing a cone at an angle that is not perpendicular to the base.
  • It is a closed curve with a distinct shape, resembling an elongated or flattened circle.
  • An ellipse has two foci (plural of “focus”) located inside the ellipse, which are not necessarily at its center.
  • The major axis is the longest diameter of the ellipse, passing through both foci. The minor axis is the shortest diameter, perpendicular to the major axis.
  • The distance between the center of the ellipse and either focus is called the semi-major axis (a).
  • The distance between the center and a point on the ellipse along the minor axis is the semi-minor axis (b).
  • The sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis (2a).
  • The eccentricity (e) of an ellipse measures how elongated it is. It is defined as the ratio of the distance between the center and one focus to the length of the semi-major axis (e = c/a, where c is the distance from the center to a focus).
  • The equation of an ellipse in standard form is (x^2/a^2) + (y^2/b^2) = 1, where (a > b).
  • In an ellipse, the sum of the distances from any point on the curve to the two foci is always less than or equal to the length of the major axis. This property is known as the triangle inequality.
  • Ellipses are commonly encountered in mathematics, physics, and engineering, particularly in planetary orbits, optics, and antenna design.

1. Equation of an Ellipse in Standard Form:
The equation of an ellipse in standard form with its center at the origin (0,0) is given by:
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
Where:
– (a, b) are the lengths of the semi-major axis and semi-minor axis, respectively.
– The major axis is along the x-axis, and the minor axis is along the y-axis.

2. Equation of an Ellipse with Center at (h, k):
If the ellipse is centered at (h, k), the standard form of the equation becomes:
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1

3. Eccentricity (e):
Eccentricity measures how elongated an ellipse is. It is calculated as:
e = \sqrt{1 - \frac{b^2}{a^2}}
Where e is the eccentricity, a is the length of the semi-major axis, and b is the length of the semi-minor axis.

4. Foci of an Ellipse:
The coordinates of the foci (F1 and F2) of an ellipse centered at (h, k) are given by:
F1: (h - ae, k)
F2: (h + ae, k)

5. Length of the Major and Minor Axes:
– The length of the major axis (2a) is twice the length of the semi-major axis: 2a = 2 \times \text{semi-major axis}.
– The length of the minor axis (2b) is twice the length of the semi-minor axis: 2b = 2 \times \text{semi-minor axis}.

6. Area of an Ellipse:
The area (A) of an ellipse is given by:
A = \pi ab
Where a is the length of the semi-major axis and b is the length of the semi-minor axis.

7. Perimeter (Circumference) of an Ellipse:
Calculating the exact perimeter of an ellipse requires special functions. An approximate formula for the circumference (C) is:
C \approx 2\pi \sqrt{\frac{a^2 + b^2}{2}}

8. Parametric Equations of an Ellipse:
Ellipses can also be described using parametric equations:
x = a \cos(\theta)
y = b \sin(\theta)
Where 0 \leq \theta \leq 2\pi is the parameter that sweeps around the ellipse.