Problem 38 - Algebra and Trigonometry

Question

\(37-40\) An equation and its graph are given. Find the \(x\) - and \(y\) -intercepts. $$ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

X-intercepts: (3, 0) and (-3, 0); Y-intercepts: (0, 2) and (0, -2).

Step 1: Understand the Equation

The given equation is \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), which represents an ellipse. The standard form for an ellipse is \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). In this case, \( a^{2} = 9 \) and \( b^{2} = 4 \), so the semi-major axis and semi-minor axis are 3 and 2 respectively.

Step 2: Find the X-intercepts

To find the \( x \)-intercepts, set \( y = 0 \) and solve for \( x \):\[ \frac{x^{2}}{9} + \frac{0^{2}}{4} = 1 \\frac{x^{2}}{9} = 1 \x^{2} = 9 \x = \pm 3 \]Thus, the \( x \)-intercepts are at \( (3, 0) \) and \( (-3, 0) \).

Step 3: Find the Y-intercepts

To find the \( y \)-intercepts, set \( x = 0 \) and solve for \( y \):\[ \frac{0^{2}}{9} + \frac{y^{2}}{4} = 1 \\frac{y^{2}}{4} = 1 \y^{2} = 4 \y = \pm 2 \]Thus, the \( y \)-intercepts are at \( (0, 2) \) and \( (0, -2) \).

Key Concepts

Conic Sections

Conic sections are fundamental concepts in geometry and mathematics, often representing curves formed by intersecting a plane with a cone. These sections include different types of curves namely circles, ellipses, parabolas, and hyperbolas. Each type has unique properties.
In this context, focusing on an ellipse, the curve can be defined as a set of points for which the sum of the distances to two fixed points (foci) remains constant. Ellipses have an oval shape and are characterized by two axes: the longer one is the major axis, and the shorter one is the minor axis. These axes are perpendicular to one another and intersect at the ellipse's center.

• The equation of an ellipse in standard form is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes.
• When \( a = b \), the ellipse becomes a circle.
• In an equation like \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), \( a^2 = 9 \) (implying \( a = 3 \)) and \( b^2 = 4 \) (implying \( b = 2 \)), hence it represents an ellipse with a semi-major axis of 3 and a semi-minor axis of 2.

X-Intercepts

The x-intercepts of a graph are the points where it intersects the x-axis. To find these points, set \( y = 0 \) in the equation and solve for \( x \).
In the equation of an ellipse, \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), the process is as follows:

• Substitute \( y = 0 \): \( \frac{x^2}{9} + \frac{0^2}{4} = 1 \), simplifying to \( \frac{x^2}{9} = 1 \).
• Multiply both sides by 9: \( x^2 = 9 \).
• Take the square root of both sides to solve for \( x \): \( x = \pm 3 \).

Thus, the x-intercepts are the points \( (3, 0) \) and \( (-3, 0) \). These points indicate where the ellipse crosses the x-axis. Understanding x-intercepts is crucial as they help in graphing and analyzing the behavior of curves and functions.

Y-Intercepts

Y-intercepts are the points where a graph crosses the y-axis; this occurs when \( x = 0 \). For the equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), finding the y-intercepts involves solving the equation by setting \( x = 0 \).
Here's the method:

• Substitute \( x = 0 \): \( \frac{0^2}{9} + \frac{y^2}{4} = 1 \), thereby simplifying to \( \frac{y^2}{4} = 1 \).
• Multiply both sides by 4: \( y^2 = 4 \).
• Take the square root of both sides to find \( y \): \( y = \pm 2 \).

This calculation reveals that the y-intercepts are \( (0, 2) \) and \( (0, -2) \). These intercepts are vital for sketching the graph of the ellipse and understanding where it interacts with the y-axis.