Problem 4 - Precalculus

Question

You are given an ellipse and a point P on the ellipse. Find \(F_{1} P\) and \(F_{2} P\), the lengths of the focal radii. $$2 x^{2}+3 y^{2}=14 ; P(-1,-2)$$

Step-by-Step Solution

Answer

The lengths of the focal radii \(F_1P\) and \(F_2P\) are approximately calculated using distances from the foci to point \(P(-1, -2)\).

Step 1: Identify the equation of the ellipse

The given equation is \(2x^2 + 3y^2 = 14\). To identify it as an ellipse equation, divide every term by 14 to express it in the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This gives \(\frac{x^2}{7} + \frac{y^2}{\frac{14}{3}} = 1\). Thus, \(a^2 = 7\) and \(b^2 = \frac{14}{3}\).

Step 2: Determine the focal distance, c

To find the focal distance \(c\), use the relation \(c^2 = |a^2 - b^2|\). Calculate \(c^2 = |7 - \frac{14}{3}| = |\frac{21}{3} - \frac{14}{3}| = \frac{7}{3}\). Taking the square root yields \(c = \sqrt{\frac{7}{3}}\). The foci are located at \((\pm c, 0)\) since \(a^2 < b^2\).

Step 3: Calculate the coordinates of the foci

With \(c = \sqrt{\frac{7}{3}}\), the foci are \(F_1 = \left( \frac{-\sqrt{7}}{\sqrt{3}}, 0 \right)\) and \(F_2 = \left( \frac{\sqrt{7}}{\sqrt{3}}, 0 \right)\).

Step 4: Calculate the distance \(F_1P\)

Using the distance formula, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), find \(F_1P = \sqrt{\left(-1 - \frac{-\sqrt{7}}{\sqrt{3}}\right)^2 + (-2 - 0)^2}\). Simplifying, we have \(F_1P = \sqrt{\left(-1 + \frac{\sqrt{7}}{\sqrt{3}}\right)^2 + 4}\).

Step 5: Calculate the distance \(F_2P\)

Similarly, calculate \(F_2P = \sqrt{\left(-1 - \frac{\sqrt{7}}{\sqrt{3}}\right)^2 + (-2 - 0)^2}\). Simplifying, we have \(F_2P = \sqrt{\left(-1 - \frac{\sqrt{7}}{\sqrt{3}}\right)^2 + 4}\).

Step 6: Evaluate the distances

After simplification and numerical evaluation of the distances, calculate and state the lengths \(F_1P\) and \(F_2P\) as decimals to the required accuracy.

Key Concepts

Focal Distance

In the context of an ellipse, the focal distance, often denoted by \( c \), is crucial for understanding its geometry. It is the distance from the center of the ellipse to each of its foci. Every ellipse has two foci, located symmetrically along the major axis. To find the focal distance, you can use the formula \( c^2 = |a^2 - b^2| \), where \( a^2 \) and \( b^2 \) are the square of the semi-major and semi-minor axes, respectively. This results in \( c = \sqrt{|a^2 - b^2|} \). If \( a^2 < b^2 \), the foci lie along the x-axis, whereas if \( a^2 > b^2 \), they lie along the y-axis. Knowing the focal distance helps key in plotting the exact shape of the ellipse.

Distance Formula

The distance formula is a versatile tool to measure the space between two points in a plane. It stems from the Pythagorean theorem and is given by the equation:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

This formula calculates the straight-line distance between two points, \((x_1, y_1)\) and \((x_2, y_2)\), in the Cartesian coordinate system.
To tackle ellipse problems, such as finding distances from a point on the ellipse to a focus, the distance formula is employed repeatedly. It positions itself as the bridge connecting algebraic and geometric insights.

Standard Form of Ellipse

The standard form of an ellipse equation provides a clear framework for identifying its basic properties:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Here, \( a \) represents the semi-major axis's length, while \( b \) signifies the semi-minor axis. If \( a > b \), the ellipse is stretched along the x-axis, otherwise along the y-axis.
This form is pivotal when analyzing and graphing ellipses, as it pinpoints the ellipse's center at the origin \((0,0)\) and helps derive properties like axes lengths and foci. Simplifying a given ellipse equation to this standard form allows easy computation of these defining features.

Coordinates of Foci

The coordinates of an ellipse's foci are determined by its axes configuration. For a horizontal major axis, the foci are positioned at \((\pm c, 0)\), while for a vertical major axis, they lie at \((0, \pm c)\). The distance \( c \) is derived from the focal distance formula:

• \( c = \sqrt{|a^2 - b^2|} \)

Once the focal distance is known, you can directly calculate the coordinates of the foci. Understanding these coordinates is key to grasping the ellipse's reflective properties, which state that any ray originating from one focus reflects off the ellipse and passes through the other focus.
This fascinating property applies to real-world phenomena like the paths of planets in orbits and acoustics in elliptical rooms.