Problem 16 - Analytical Geometry: 2D and 3D

Question

Find the latus rectum of the hyperbola \(4 x-9 y^{2}=36\).

Step-by-Step Solution

Answer

Answer: The latus rectum of the hyperbola is \(\frac{8}{3}\).

Step 1: Rewrite the equation in standard form

To write the equation of the hyperbola in the standard form, we must express it as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where a and b are the semi-major and semi-minor axes respectively. Given equation is \(4x - 9y^2 = 36\). Divide the equation by 36 to get:$$\frac{x^2}{9} - \frac{y^2}{4} = 1$$Now, the equation is in the standard form.

Step 2: Identify the values of a and b

In the standard form of the hyperbola equation, we can observe that:$$a^2 = 9 \Rightarrow a = 3$$$$b^2 = 4 \Rightarrow b = 2$$So, the semi-major axis a is 3 and the semi-minor axis b is 2.

Step 3: Calculate the latus rectum

The latus rectum of a hyperbola is given by the formula:$$Latus\ Rectum = \frac{2b^2}{a}$$Substitute the values of a and b in this formula:$$Latus\ Rectum = \frac{2\times (2)^2}{3}$$Simplify the expression:$$Latus\ Rectum = \frac{8}{3}$$ So, the latus rectum of the hyperbola \(4x - 9y^2 = 36\) is \(\frac{8}{3}\).

Key Concepts

Latus Rectum

The term "latus rectum" refers to an important geometric feature of conic sections. Here, we focus on its role within hyperbolas. Essentially, the latus rectum is a line segment that is perpendicular to the transverse axis of a hyperbola and passes through the foci. It is a symmetrical part of the hyperbola.

Understanding how to calculate the latus rectum is crucial as it provides insight into the properties of the hyperbola. The formula used to find the latus rectum of a hyperbola is:

• \(\text{Latus Rectum} = \frac{2b^2}{a}\)

This formula means we square the length of the semi-minor axis, multiply it by 2, then divide by the length of the semi-major axis. This detail signifies not only the dimensions but also how these attributes of the hyperbola interact geometrically.

Standard Form of Hyperbola

Getting the equation of a hyperbola into the standard form is the crucial first step in understanding its shape and properties.

The standard form of a hyperbola is given by:

• Horizontal transverse axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Vertical transverse axis: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

In the given exercise, the equation \(4x - 9y^2 = 36\) needed to be modified to fit the first format. The rewriting process involves algebraic manipulation, including division and rearranging terms. By dividing the equation by 36, we transformed it into:

• \(\frac{x^2}{9} - \frac{y^2}{4} = 1\)

In this standard form, you readily identify the axes lengths, which are crucial for calculating further properties like the latus rectum.

Semi-Major Axis

The semi-major axis is one of the key parameters of a hyperbola. It lies along the main axis of symmetry and determines the "width” of the hyperbola spread between its vertices.

In the context of the given exercise, the semi-major axis comes from the term associated with the variable having a minus sign in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Here, this indicates that the axis is along the x-direction.

From the equation \(\frac{x^2}{9} - \frac{y^2}{4} = 1\), we derived:

• \(a^2 = 9\)

Thus, the length of the semi-major axis is calculated as \(a = 3\). This length is part of essential calculations such as the latus rectum.

Semi-Minor Axis

The semi-minor axis is the perpendicular distance from the center of the hyperbola to a point called the co-vertex. It is crucial for determining the hyperbola's vertical spread.

In a hyperbola's standard equation, the term that does not have a negative sign defines the length of the semi-minor axis. For the exercise in question, once the equation is in its standard form

• \(\frac{x^2}{9} - \frac{y^2}{4} = 1\)

we deduced that:

• \(b^2 = 4\)

Therefore, the semi-minor axis is \(b = 2\). This value plays a fundamental role in not just deriving geometric properties, but also in understanding the path and nature of the hyperbola.