Find the length of the latus rectum of the parabola whose focus is at (2, 3) and directrix is the line x − 4y + 3 = 0.

To find the length of the latus rectum of the parabola with a focus at (2, 3) and a directrix given by the equation x − 4y + 3 = 0, we can follow these steps:

Understanding the Components

The latus rectum is a line segment that passes through the focus of the parabola and is perpendicular to the axis of symmetry. Its length can be calculated using the formula:

Length of Latus Rectum = 4p

where p is the distance from the focus to the directrix.

Finding the Distance p

First, we need to determine the distance from the focus (2, 3) to the directrix. The directrix can be rewritten in the standard form:

For the line x − 4y + 3 = 0, we can express it as:

Ax + By + C = 0

where A = 1, B = -4, and C = 3.

Using the Distance Formula

The distance d from a point (x₀, y₀) to a line Ax + By + C = 0 is given by:

d = |Ax₀ + By₀ + C| / √(A² + B²)

Substituting the values:

• A = 1
• B = -4
• C = 3
• x₀ = 2
• y₀ = 3

We calculate:

d = |1(2) - 4(3) + 3| / √(1² + (-4)²)

This simplifies to:

d = |2 - 12 + 3| / √(1 + 16)

Which further simplifies to:

d = | -7 | / √17

Thus, d = 7 / √17.

Calculating the Length of the Latus Rectum

Now that we have p, we can find the length of the latus rectum:

Length = 4p = 4(7 / √17)

This results in:

Length = 28 / √17

To summarize, the length of the latus rectum of the given parabola is 28 / √17.