Question

Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum 2 \(a .\) Derive the equation of an ellipse. Assume the two fixed points are on the \(x\) -axis equidistant from the origin.

Short answer

Answer: The equation of an ellipse in standard form is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(b^2 = a^2 - c^2\), a is the semi-major axis, and c is the distance from the origin to each focus.

Step-by-step solution

Define the coordinates of the foci and a point on the ellipse

Let's denote the two foci as F1(-c, 0) and F2(c, 0), where c is the distance from the origin along the x-axis. Now, consider a point P(x, y) on the ellipse such that the sum of the distances PF1 and PF2 is constant, say 2a, where a > c.

Apply the distance formula to find PF1 and PF2

To find the distances PF1 and PF2, we can use the distance formula: PF1 = \(\sqrt{(x + c)^2 + y^2}\) PF2 = \(\sqrt{(x - c)^2 + y^2}\)

Set up the equation representing the sum of PF1 and PF2

According to the definition of an ellipse, the sum of PF1 and PF2 is equal to the constant 2a: \(\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a\)

Manipulate the equation to obtain the ellipse equation

Squaring both sides: \([(x + c)^2 + y^2] + 2\sqrt{((x + c)^2 + y^2)((x - c)^2 + y^2)} + [(x - c)^2 + y^2] = (2a)^2\) It's essential to eliminate the square root term. Subtract \((x^2 + y^2)\) from both sides: \(2\sqrt{((x + c)^2 + y^2)((x - c)^2 + y^2)} = 4a^2 - 4c^2\) Divide both sides by 2: \(\sqrt{((x + c)^2 + y^2)((x - c)^2 + y^2)} = 2a^2 - 2c^2\) Now, square both sides again: \(((x + c)^2 + y^2)((x - c)^2 + y^2) = (2a^2 - 2c^2)^2\) Expanding both sides and simplifying the terms, you will get: \((x^2 + 2cx + c^2 + y^2)(x^2 - 2cx + c^2 + y^2) = 4a^4 - 8a^2c^2 + 4c^4\) Expanding again and simplifying the terms, the equation will reduce to: \(-c^2x^2 - c^2y^2 + a^4 - 2a^2c^2 + c^4 = 0\) Finally, we'll rearrange the equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(b^2 = a^2 - c^2\) This is the equation of an ellipse in standard form.