6.5 Q-5

Question Statement

Define the latus rectum of an ellipse and prove that the length of the latus rectum for the ellipse

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$

is $\frac{2b^{2}}{a}$.

---

Background and Explanation

The latus rectum of an ellipse is the chord passing through one of the foci and perpendicular to the major axis. It is a key feature of the ellipse as it helps in understanding the geometrical properties related to the focal points.

For the standard ellipse equation:

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$$

where $a > b$:

• $a$ represents the semi-major axis length.
• $b$ represents the semi-minor axis length.
• The distance from the center to a focus is given by $c = \sqrt{a^{2} - b^{2}}$.

---

Solution

Step 1: Establish the coordinates of the latus rectum

The latus rectum is a vertical line passing through the foci. Let $(c, y)$ be a point on the ellipse where $c$ is the distance of the focus from the center. Substituting $x = c$ into the equation of the ellipse:

$$\frac{c^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.$$

Step 2: Solve for $y^{2}$

Reorganize the equation to isolate $y^{2}$:

$$\frac{y^{2}}{b^{2}} = 1 - \frac{c^{2}}{a^{2}}.$$

Since $c^{2} = a^{2} - b^{2}$, substitute this into the equation:

$$\frac{y^{2}}{b^{2}} = \frac{a^{2} - c^{2}}{a^{2}} = \frac{b^{2}}{a^{2}}.$$

Multiply through by $b^{2}$:

$$y^{2} = \frac{b^{4}}{a^{2}}.$$

Thus, the coordinates of the points on the latus rectum are:

$$(c, \pm \frac{b^{2}}{a}).$$

Step 3: Length of the latus rectum

The total length of the latus rectum is the distance between the points:

$$\text{Length of latus rectum} = 2 \left| \frac{b^{2}}{a} \right| = \frac{2b^{2}}{a}.$$

---

Key Formulas or Methods Used

1. Standard Equation of an Ellipse:

$$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.$$

2. Relationship Between Semi-Major and Semi-Minor Axes:

$$c^{2} = a^{2} - b^{2},$$

where $c$ is the focal distance. 3. Length of the Latus Rectum: Derived as $\frac{2b^{2}}{a}$.

---

Summary of Steps

1. Start with the standard ellipse equation:

$$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.$$

2. Substitute $x = c$, where $c = \sqrt{a^{2} - b^{2}}$, into the equation to solve for $y^{2}$.
3. Calculate the total length of the latus rectum as:

$$\text{Length} = 2 \cdot \frac{b^{2}}{a}.$$

---