Understanding the Equation of an Ellipse with Given Focus and Point

In the realm of analytic geometry, the equation of an ellipse is a fundamental concept that can be derived from a set of given data points. This article will elucidate how to find the equation of an ellipse when one focus and a point on the ellipse are provided. We will use a step-by-step approach to solve a specific problem where we need to determine the equation of an ellipse having one focus at -4,2 and passing through the point 4,6.

Step 1: Determine the Center of the Ellipse

The first step is to find the center of the ellipse. Given that one focus is at (-4, 2) and the point (4, 6) lies on the ellipse, the center of the ellipse is the midpoint between these two points.

Calculating the Center

If we denote the other focus as (c, 2), the center ( C ) of the ellipse is calculated as:

C left(frac{-4 c}{2}, 2right)

Since the point (4, 6) lies on the ellipse, we can equate the x-coordinate of the center to the x-coordinate of the given point:

frac{-4 c}{2} 4

Solving for ( c ):

-4 c 8

c 12

Thus, the coordinates of the other focus are (12, 2), and the center of the ellipse is:

C left(frac{-4 12}{2}, 2right) (4, 2)

Step 2: Calculate the Distance Between the Foci

The distance between the foci is given by:

2c 12 - (-4) 16 implies c 8

Step 3: Use the Point (4, 6) to Find a and b

Since the point (4, 6) lies on the ellipse, we can use it to find the semi-major axis ( a ) and the semi-minor axis ( b ).

Calculating the Distance from the Center to the Point

The distance from the center (4, 2) to the point (4, 6) is:

d sqrt{(4 - 4)^2 (6 - 2)^2} sqrt{0 16} 4

This distance ( d ) is the semi-major axis ( a ) and is denoted as:

a 4

The relationship between ( a ), ( b ), and ( c ) for an ellipse is given by:

c^2 a^2 - b^2

Calculating b

Substituting the values of ( a ) and ( c ):

8^2 4^2 - b^2

64 16 - b^2

b^2 -48

Since ( b^2 -48 ) is not possible (a real number cannot be negative under the square root), it indicates that the ellipse degenerates into a line segment between the two foci.

Final Equation

Since the ellipse degenerates into a line segment, it does not represent a typical ellipse but rather the line segment joining the two foci. The final equation is effectively represented by the coordinates of the foci and the center:

x - 4 0

Thus, the degenerative form of the ellipse is:

x - 4 0

In conclusion, the given problem indicates that the ellipse collapses into a vertical line segment at the x-coordinate of the center, which is 4, between the points (-4, 2) and (12, 2).