Question

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of an ellipse centered at the origin is \(2 b^{2} / a=2 b \sqrt{1-e^{2}}\)

Short answer

Question: Prove that the length of the latus rectum of an ellipse centered at the origin is given by the formula \(2 b^2 / a = 2 b \sqrt{1 - e^2}\), where \(a\) and \(b\) are the semi-major and semi-minor axes respectively, and \(e\) is the eccentricity of the ellipse. Answer: To prove that the length of the latus rectum of an ellipse centered at the origin is given by the formula \(2 b^2 / a = 2 b \sqrt{1 - e^2}\), we first find the coordinates of the endpoints of the latus rectum using the equation of the ellipse, the eccentricity definition, and the formula for the foci. Then, we use the distance formula to find the length of the latus rectum, which results in \(2b\sqrt{1-e^2}\).

Step-by-step solution

Equation of the ellipse

The ellipse centered at the origin with semi-major axis \(a\), semi-minor axis \(b\) can be represented by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Eccentricity of the ellipse

The eccentricity \(e\) of an ellipse is defined as \(e = \sqrt{1 - \frac{b^2}{a^2}}\).

Foci of the ellipse

The foci of an ellipse can be found using the formula \(c = ae\), where \(c\) is the distance from the center of the ellipse to each focus. The coordinates of the foci are \((\pm c, 0)\).

Coordinates of endpoints of the latus rectum

Since the latus rectum passes through the focus and is perpendicular to the major axis, its endpoints will have the same \(y\)-coordinates. Using the equation of the ellipse, we can find these \(y\)-coordinates by substituting the \(x\)-coordinate of one of the foci, \(c\): \(\frac{c^2}{a^2} + \frac{y^2}{b^2} = 1\) Solving for \(y^2\) gives \(y^2 = b^2 (1 - \frac{c^2}{a^2})\). Since \(c = ae\), we have \(y^2 = b^2 (1 - e^2)\). The coordinates of the endpoints of the latus rectum are then \((c, \pm b\sqrt{1-e^2})\).

Length of the latus rectum

Using the distance formula, the length of the latus rectum is: \(2b\sqrt{1-e^2}\). This completes the proof.