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Chapter 2: Problem 99

Suppose \(a>b>0\). Find a formula in terms of \(x\) for the distance from a typical point \((x, y)\) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) to the point \(\left(\sqrt{a^{2}-b^{2}}, 0\right) .\)

Short Answer

Expert verified
The distance formula between a point \((x, y)\) on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and the point \(\left(\sqrt{a^2 - b^2}, 0\right)\) is given by: \(D(x, y) = \sqrt{(\sqrt{a^2 - b^2} - x)^2 + y^2}\)

Step by step solution

01

Identify the Given Points

We are given the points: 1. A point \((x, y)\) lying on the ellipse, which satisfies the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). 2. A fixed point \(\left(\sqrt{a^2 - b^2}, 0\right)\).
02

Find the Distance Formula

To calculate the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the distance formula: \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
03

Apply the Distance Formula to the Given Points

Now, apply this formula to the coordinates of our points \((x, y)\) and \(\left(\sqrt{a^2 - b^2}, 0\right)\): \(D(x, y) = \sqrt{(\sqrt{a^2 - b^2} - x)^2 + (0 - y)^2}\)
04

Simplify the Expression

Finally, simplify the distance expression: \(D(x, y) = \sqrt{(\sqrt{a^2 - b^2} - x)^2 + y^2}\) This is the formula for the distance between a point \((x, y)\) on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) and the point \(\left(\sqrt{a^2 - b^2}, 0\right)\).

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