Problem 69 - Algebra 2

Question

The point \(A(-10,0)\) is on the ellipse with equation \(\frac{x^{2}}{100}+\frac{y^{2}}{64}=1 .\) What is the sum of the distances \(A F_{1}+A F_{2},\) where \(F_{1}\) and \(F_{2}\) are toci? \(\begin{array}{llll}{\text { A. } 10} & {\text { B. } 12} & {\text { C. } 14} & {\text { D. } 20}\end{array}\)

Step-by-Step Solution

Answer

The sum of the distances \(A F_{1}+A F_{2} is 20 units. Therefore, the correct answer is D. 20.

Step 1 Identify the major axis a

From the given equation of the ellipse, the major axis can be identified as 2a, where a is the square root of the largest denominator in the equation. Hence a = sqrt(100) = 10. Therefore, the major axis is 2a = 2*10 = 20 units.

Step 2 Apply property of ellipse

The property of an ellipse states that the sum of the distances from any point on the ellipse to the two foci is constant, and is equal to the length of the major axis.

Step 3 Calculate the sum of distances

Since the length of the major axis is 20 units, this will be the sum of the distances from the point A to the two foci (F1 and F2).

Key Concepts

Equation of Ellipses

The equation of an ellipse is a mathematical expression that describes the set of all points whose sum of distances to two fixed points, called foci, is constant. This equation can generally be written in the standard form:

• Horizontal orientation: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
• Vertical orientation: \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \)

Here, \((h, k)\) represents the center of the ellipse.
The terms \(a\) and \(b\) are the semi-major and semi-minor axes respectively, with \(a\) being the larger.
In the given problem, the equation of the ellipse is \(\frac{x^2}{100} + \frac{y^2}{64} = 1\), indicating a horizontal orientation because \(a^2 = 100\) is greater than \(b^2 = 64\).
This tells us the semi-major axis \(a\) is 10 and the semi-minor axis \(b\) is 8.

Foci of Ellipses

The foci of an ellipse are two distinct points located along the major axis. The unique property of these points is that the sum of the distances from any point on the ellipse to the foci is constant.
To find the foci of an ellipse, we need to calculate the distance from the center to each focus. This distance is denoted as \(c\) and can be determined using the equation \(c = \sqrt{a^2 - b^2}\).

• For our ellipse, where \(a = 10\) and \(b = 8\):
• \(c = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6\)

Thus, the foci are located at the points \((-6,0)\) and \((6,0)\), considering the center is at the origin \((0,0)\). These foci play a crucial role in defining the shape and properties of the ellipse.

Properties of Ellipses

Ellipses have fascinating properties that distinguish them from other conic shapes. A standout property is that for any ellipse, the sum of the distances from any point on the ellipse to the two foci is always equal to the length of the major axis. This is a fundamental characteristic, and in our problem, it means:
When the major axis length is calculated, it literally is the sum of the distances from any point on the ellipse to each focus. Here, the major axis length is calculated in Step 1 as 20 units.

• The given formula results in a constant, which for the ellipse in question, is 20.
• Therefore, the sum of the distances from point \(A(-10,0)\) on the ellipse to foci \(F_1\) and \(F_2\) is consistently 20 units.

This property not only simplifies many calculations involving ellipses but also reflects its invariant geometric nature.