Solving the Ellipse Puzzle: Equation in Standard Form

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Solving the Ellipse Puzzle: Equation in Standard Form
Finding the Equation of an Ellipse: Step-by-Step Guide

Finding the Equation of an Ellipse: Step-by-Step Guide

Ellipses are intriguing geometric shapes with distinct properties. Let's unravel the mystery of an ellipse with a center at (6,5), a horizontal major axis of length 8, and passing through the point (8,3). We'll find its equation in standard form using a systematic approach:

Step 1: Understand the Given Information

Center: \((h, k) = (6, 5)\)
Horizontal Major Axis Length: \(2a = 8 \Rightarrow a = 4\)
Point on the Ellipse: \((8, 3)\)

Step 2: Use the Ellipse Equation for a Horizontal Major Axis

The standard form of an ellipse with a horizontal major axis and center \((h, k)\) is:

\[ \frac{{(x - h)^2}}{{a^2}} + \frac{{(y - k)^2}}{{b^2}} = 1 \]

Step 3: Substitute Values and Solve for \(b\)

Substitute the values into the ellipse equation:

\[ \frac{{(x - 6)^2}}{{4^2}} + \frac{{(y - 5)^2}}{{b^2}} = 1 \]

Substitute the point (8,3):

\[ \frac{{(8 - 6)^2}}{{4^2}} + \frac{{(3 - 5)^2}}{{b^2}} = 1 \] \[ \frac{{4}}{{16}} + \frac{{4}}{{b^2}} = 1 \] \[ \frac{{1}}{{4}} + \frac{{4}}{{b^2}} = 1 \]

Solve for \(b^2\):

\[ \frac{{4}}{{b^2}} = \frac{{3}}{{4}} \] \[ b^2 = \frac{{16}}{{3}} \implies b = \frac {4}{\sqrt3} = \frac{4 \sqrt 3}{3} \]

Step 4: Write the Standard Form Equation

The equation of the ellipse in standard form is:

\[ \frac {(x - 6)^2}{4^2} + \frac{(y - 5)^2}{[{\frac{4 \sqrt 3}{3}}]^2} = 1 \]

And there you have it! The equation of the given ellipse in standard form.

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