Questions: What is (are) the value(s) of x at the vertex (vertices) of the shape defined by the equation x^2/64 + y^2/36 = 1? (Separate multiple values of x with a comma.)
Transcript text: What is (are) the value(s) of $x$ at the vertex (vertices) of the shape defined by the equation $\frac{x^{2}}{64}+\frac{y^{2}}{36}=1$ ? (Separate multiple values of $x$ with a comma.)
Solution
Solution Steps
Step 1: Identify the Equation of the Ellipse
The given equation is 64x2+36y2=1. This is the standard form of an ellipse centered at the origin (0,0) with semi-major and semi-minor axes.
Step 2: Determine the Semi-Major and Semi-Minor Axes
In the standard form a2x2+b2y2=1, the values a2=64 and b2=36 are given. The semi-major axis is along the x-axis because a2>b2.
Step 3: Calculate the Length of the Semi-Major Axis
The length of the semi-major axis is given by a=64=8.
Step 4: Identify the x-Coordinates of the Vertices
The vertices of the ellipse along the x-axis are located at (±a,0). Therefore, the x-coordinates of the vertices are x=−8 and x=8.