Math Problem Statement

Given the ellipse
( x + 5 ) 2 25 + ( y − 3 ) 2 9

1

Find the center point:

Give the vertices:

Give the endpoints of the minor axis:

Draw the graph of the ellipse.

Solution

Here is the graph of the ellipse with the center, vertices, and endpoints of the minor axis highlighted:

  • The center is at (5,3)(-5, 3).
  • The vertices are at (0,3)(0, 3) and (10,3)(-10, 3).
  • The endpoints of the minor axis are at (5,6)(-5, 6) and (5,0)(-5, 0).

Would you like more details on any of these points or further explanation?

Here are five related questions you might find helpful:

  1. How do you determine the eccentricity of an ellipse?
  2. What is the formula for the area of an ellipse?
  3. How does changing the values of aa and bb affect the shape of the ellipse?
  4. How can we rotate an ellipse around its center?
  5. What are some applications of ellipses in physics or astronomy?

Tip: The length of the major and minor axes are twice the values of aa and bb respectively.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Ellipses
Conic Sections

Formulas

Standard form of an ellipse: (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Center of an ellipse: (h, k)
Vertices of an ellipse: (h ± a, k)
Endpoints of the minor axis: (h, k ± b)

Theorems

The Pythagorean Theorem (for deriving the relation between axes and foci)
Properties of ellipses in conic sections

Suitable Grade Level

Grades 10-12