Question

How long is the latus rectum (chord through the focus perpendicular to the major axis) for the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1 ?\)

Short answer

The latus rectum length is \(\frac{2b^2}{a}\).

Step-by-step solution

Understand the Components of the Ellipse

An ellipse is given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. We need to determine the length of the latus rectum, a line segment perpendicular to the major axis of the ellipse that passes through one of the foci.

Identify the Focus of the Ellipse

For an ellipse in the form provided, the foci are located at \((\pm c, 0)\) for a horizontal major axis, where \(c = \sqrt{a^2 - b^2}\). The focus used here is \((c, 0)\), assuming \(a > b\) so the major axis is horizontal.

Use the Formula for Latus Rectum

The length of the latus rectum for an ellipse is given by the formula \(\frac{2b^2}{a}\). This formula ensures the perpendicular chord through the focus intersects the ellipse, maintaining balance relative to the ellipse's central symmetry.

Calculate the Length

Substitute the semi-axis lengths \(a\) and \(b\) into the formula. The length of the latus rectum is therefore \(\frac{2b^2}{a}\).

Key concepts

Latus Rectum

The term *latus rectum* might sound complex but it is a simple concept in ellipse calculus. The latus rectum of an ellipse is the line segment that is perpendicular to the major axis and passes through one of the foci of the ellipse. Think of it as a special bridge within the ellipse that connects points on either side, through one focal point. Its importance comes from how it helps you understand the ellipse's symmetry and structure in depth.

To find the length of this segment, there's a straightforward formula: \( \frac{2b^2}{a} \). This formula uses the values of the semi-minor axis \(b\) and the semi-major axis \(a\). This ensures that the latus rectum maintains the symmetry and balance of the ellipse while accurately spanning across one of the foci, making it a significant feature in the geometry of ellipses.

Ellipse Equation

The standard form of an ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This equation is fundamental to understanding how an ellipse is structured. Here, \(a\) represents the length of the semi-major axis, while \(b\) represents the length of the semi-minor axis. These axes help define the ellipse's width and height.

The values of \(a\) and \(b\) determine the shape of the ellipse. If \(a > b\), the ellipse is elongated along the x-axis, which means the major axis is horizontal. The opposite occurs when \(b > a\), aligning the major axis vertically along the y-axis. Understanding the ellipse equation is crucial because it provides insights into the ellipse's dimensions and orientation, which is foundational when further exploring topics like the foci or the latus rectum.

Foci of Ellipse

Inside every ellipse, there are two special points known as the foci (singular: focus). The foci are crucial in defining the curve of the ellipse because they are points where the sum of the distances from any point on the ellipse to the foci remains constant.

For the ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the foci are located at \((\pm c, 0)\). Here, \(c\), the distance from the center of the ellipse to a focus, is calculated by \(c = \sqrt{a^2 - b^2}\). This shows how interconnected \(a\), \(b\), and \(c\) are, where knowing any two can help deduce the third.

Understanding the concept of the foci is significant because it explains why the ellipse's shape is stretched in certain directions. It helps us comprehend the unique property of ellipses where light or sound emanating from one focus will converge at the other, a principle used in various scientific and engineering applications.