Question

Write the equation of the given ellipse in standard form. \(5 x^{2}+3 y^{2}=15\)

Short answer

The standard form is \(\frac{x^2}{3} + \frac{y^2}{5} = 1\).

Step-by-step solution

Understanding the Ellipse Equation

An ellipse centered at the origin with axes aligned with the coordinate axes has the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes.

Rewrite the Given Equation

Begin by rewriting the given equation, \(5x^2 + 3y^2 = 15\), to get it into the standard form. The goal is to have the right side of the equation equal to 1.

Divide by 15

Divide every term in the equation \(5x^2 + 3y^2 = 15\) by 15 to achieve this:\[ \frac{5x^2}{15} + \frac{3y^2}{15} = 1 \].

Simplify the Fractions

Simplify the fractions:\[ \frac{x^2}{3} + \frac{y^2}{5} = 1 \]. This is now in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), corresponding to an ellipse.

Key concepts

Equation of an Ellipse

The equation of an ellipse provides a mathematical representation of the shape, which is essentially an elongated circle. Just like a circle has one radius, an ellipse has two axes: a major axis and a minor axis. The general equation of an ellipse with its center at the origin and aligned with the coordinate axes is given by:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \].In this equation:

• \(a\) represents the length of the semi-major axis.
• \(b\) represents the length of the semi-minor axis.

This equation is essential as it tells us how stretched the ellipse is. By analyzing the values of \(a\) and \(b\), we can determine the shape's orientation and size.

Standard Form of an Ellipse

The standard form of an ellipse is crucial for easily identifying its geometric properties. For the equation to be in standard form, the right side must equate to 1. This makes it straightforward to visualize the dimensions of the ellipse. The process often involves manipulation, such as dividing or factoring, to bring equations into this form. For instance, with the original equation \(5x^2 + 3y^2 = 15\), we divide all terms by 15 to get:\[ \frac{5x^2}{15} + \frac{3y^2}{15} = 1 \],which simplifies to:\[ \frac{x^2}{3} + \frac{y^2}{5} = 1 \].By having the equation in this form, it becomes easier to extract meaningful information about the ellipse’s dimensions, helping us plot and interpret it correctly.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse and extends from the center to the farthest point on the perimeter. In the standard form equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the value of \(a\) corresponds to the semi-major axis. Depending on which term (\(x^2\) or \(y^2\)) has the larger denominator, \(a\) will be that value. As seen in our simplified equation, \(\frac{x^2}{3} + \frac{y^2}{5} = 1\), the larger denominator is 5. Hence:

• The length of the semi-major axis is \(\sqrt{5}\).

This axis determines the longest distance across the ellipse, which significantly influences its overall appearance and orientation.

Semi-Minor Axis

The semi-minor axis is the shortest radius of an ellipse, stretching from the center to the closest point on the perimeter. In the standard equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the value of \(b\) represents the semi-minor axis. For the equation we derived, \(\frac{x^2}{3} + \frac{y^2}{5} = 1\), the smaller denominator is 3. Consequently:

• The length of the semi-minor axis is \(\sqrt{3}\).

Understanding the semi-minor axis helps visualize the height and the y-axis spread of the ellipse, which is crucial for determining how compact the ellipse appears.