Equation of Ellipse in Other Forms

Equation of Ellipse in Other Forms

General Equation of an Ellipse:

The standard equation of an ellipse represents an ellipse centered at the origin (0,0). However, ellipses might not always be centered at the origin. The general equation of an ellipse with its center at (h,k) can be expressed as:

(xh)2a2+(yk)2b2=1

Where:

  • (h,k) denotes the coordinates of the center.
  • a is the length of the semi-major axis.
  • b is the length of the semi-minor axis.

Equation of an Ellipse in Parametric Form:

The parametric equations of an ellipse allow the representation of points on the ellipse using parameters t (usually angle or time). The parametric form of an ellipse with center (h,k), semi-major axis a, and semi-minor axis b is given by:

x=h+acos(t)   y=k+bsin(t)

Where t varies over the interval [0,2π] for a complete ellipse.

Rotated Ellipse Equation:

An ellipse can be rotated at an angle θ with respect to the x-axis. The equation of a rotated ellipse, centered at (h,k), can be derived using rotation formulas. The general form of a rotated ellipse is:

(xh)2a2cos2(θ)+(yk)2b2sin2(θ)=1

Where a and b are the semi-major and semi-minor axes lengths respectively, and θ is the angle of rotation.

Cartesian Form from Parametric Equations:

The Cartesian form of an ellipse can be derived from its parametric equations by eliminating the parameter t. For an ellipse centered at the origin (0,0) with semi-major axis a and semi-minor axis b, the Cartesian equation is:

x2a2+y2b2=1

General Form of Ellipse:

 Ax2+By2+Cx+Dy+E=0

    • This equation represents an ellipse that might not necessarily have its center at the origin (0,0).
    • It's a more generalized form of the ellipse equation.

Polar Equation of Ellipse:

 r(θ)=aba2sin2(θ)+b2cos2(θ)

  • Describes the ellipse in polar coordinates with the center at the origin.
  • r represents the distance from the center to a point on the ellipse at an angle θ.

Canonical Form of Ellipse:

 x2a2+y2b2=1

    • Represents an ellipse centered at the origin with the major axis along the x-axis and minor axis along the y-axis.
    • a and b represent the semi-major and semi-minor axes, respectively.

Transformations and Rotations:

  1. Horizontal and Vertical Translations:

    • Adding or subtracting constants to x and y terms in the standard equation shifts the ellipse horizontally or vertically.
  2. Rotation of Axes:

    • An ellipse can be rotated by an angle θ by applying rotation formulas to x and y in the standard equation.
  3. Completing the Square:

    • The standard form can be obtained from the general form by completing the square for both x and y terms.

Example 1: General Form of Ellipse

Given: The equation of an ellipse is 3x2+4y26x+16y11=0. Find the center, semi-major and semi-minor axes, and sketch the ellipse.

Solution:

To rewrite the given equation in the standard form, complete the square for both x and y terms:

3x26x+4y2+16y=11

Completing the square for x: 3(x22x)+4y2+16y=11 3(x22x+1)3+4(y2+4y+4)16=11

 3(x1)2+4(y+2)2=30

Divide by 30 to get the standard form: (x1)210+(y+2)27.5=1

Now, we have the equation in the standard form. The center is (h,k)=(1,2), the semi-major axis is a=10, and the semi-minor axis is b=7.5.

Example 2: Parametric Equations of Ellipse

Given: The center of an ellipse is (3,2), with a semi-major axis a=4 and a semi-minor axis b=3. Find the parametric equations of the ellipse.

Solution:

The parametric equations for an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b are:

x=h+acos(θ) y=k+bsin(θ)

Substitute the given values: x=3+4cos(θ) y=2+3sin(θ)

These equations represent the ellipse in terms of parameter θ from 0 to 2π.

Example 3: Polar Equation of Ellipse

Given: An ellipse has a center at the origin, with semi-major axis a=5 and semi-minor axis b=3. Find the polar equation of the ellipse.

Solution:

The polar equation for an ellipse centered at the origin with semi-major axis a and semi-minor axis b is:

r(θ)=aba2sin2(θ)+b2cos2(θ)

Substitute the given values: r(θ)=5×325sin2(θ)+9cos2(θ)

This equation describes the ellipse in polar coordinates with respect to the angle θ.