Math Problem Statement

Identify the type of conic section given by the equation: 16x^2 + 4y^2 - 224x - 32y + 784 = 0.

Solution

To solve this, let’s analyze the given equation:

16x2+4y2224x32y+784=016x^2 + 4y^2 - 224x - 32y + 784 = 0

Steps to Identify the Conic Section:

  1. Compare the coefficients of x2x^2 and y2y^2:

    • The coefficients are 1616 and 44, both positive, which suggests it might be an ellipse.
  2. Simplify the equation into standard form: Let's complete the square for xx and yy.

    Completing the square for xx:

    16x2224x=16(x214x)16x^2 - 224x = 16(x^2 - 14x) Complete the square: x214x    (x7)249x^2 - 14x \implies (x-7)^2 - 49 Substituting back: 16(x214x)=16((x7)249)=16(x7)278416(x^2 - 14x) = 16((x-7)^2 - 49) = 16(x-7)^2 - 784

    Completing the square for yy:

    4y232y=4(y28y)4y^2 - 32y = 4(y^2 - 8y) Complete the square: y28y    (y4)216y^2 - 8y \implies (y-4)^2 - 16 Substituting back: 4(y28y)=4((y4)216)=4(y4)2644(y^2 - 8y) = 4((y-4)^2 - 16) = 4(y-4)^2 - 64

  3. Rewriting the equation: Substitute completed squares into the equation: 16(x7)2784+4(y4)264+784=016(x-7)^2 - 784 + 4(y-4)^2 - 64 + 784 = 0 Simplify: 16(x7)2+4(y4)2848+784=016(x-7)^2 + 4(y-4)^2 - 848 + 784 = 0 16(x7)2+4(y4)264=016(x-7)^2 + 4(y-4)^2 - 64 = 0 16(x7)2+4(y4)2=6416(x-7)^2 + 4(y-4)^2 = 64

  4. Divide through by 64 to standardize: 16(x7)264+4(y4)264=1\frac{16(x-7)^2}{64} + \frac{4(y-4)^2}{64} = 1 (x7)24+(y4)216=1\frac{(x-7)^2}{4} + \frac{(y-4)^2}{16} = 1

    This is the equation of an ellipse.

Final Answer:

The given equation represents an ellipse.


Do you need further clarification or details? Here are 5 related questions for further exploration:

  1. What are the major and minor axes of the ellipse derived above?
  2. How do you determine the center of a conic section from its equation?
  3. What distinguishes an ellipse from a hyperbola in their general equations?
  4. Can you explain the process of completing the square in more detail?
  5. How would this equation look if it were a circle instead of an ellipse?

Tip: Always check the signs and relative magnitudes of the squared terms’ coefficients to quickly narrow down the type of conic section.

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Math Problem Analysis

Mathematical Concepts

Conic Sections
Ellipse
Completing the Square

Formulas

Equation of ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
Completing the square

Theorems

Properties of Conic Sections

Suitable Grade Level

College-level Algebra