Question

sketch a graph of $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$

Explanation:

Step1: Identify the form of the equation

The given equation \(\frac{x^{2}}{25}+\frac{y^{2}}{9} = 1\) is in the standard form of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), where \(a^{2}=25\) (so \(a = 5\)) and \(b^{2}=9\) (so \(b = 3\)).

Step2: Find the vertices

For an ellipse of the form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), the vertices on the \(x\) - axis are \((\pm a,0)\). Substituting \(a = 5\), the vertices are \((5,0)\) and \((- 5,0)\).

Step3: Find the co - vertices

The co - vertices on the \(y\) - axis are \((0,\pm b)\). Substituting \(b = 3\), the co - vertices are \((0,3)\) and \((0,-3)\).

Step4: Sketch the ellipse

Plot the vertices \((\pm5,0)\) and co - vertices \((0,\pm3)\). Then draw a smooth curve passing through these four points to form the ellipse.

Answer:

Sketch an ellipse with vertices \((\pm5,0)\) and co - vertices \((0,\pm3)\) using the given grid.