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Find a Cartesian equation for the ellipse with the given properties

by RoRi

Find a Cartesian equation in standard form for the ellipse with center at (0,0), one focus at \left( \frac{3}{4}, 0 \right), and one vertex at (1,0).

Since the ellipse has center at (0,0), the vertices are \pm aN. Therefore, we have a = 1. Then from the given focus we have

    \[ c = \sqrt{a^2- b^2} \quad \implies \quad \frac{3}{4} = \sqrt{1 - b^2} \quad \implies \quad b = \frac{\sqrt{7}}{4}. \]

Therefore, the Cartesian equation for the ellipse is

    \[ x^2 + \frac{16y^2}{7} = 1 \quad \implies \quad 7x^2 + 16y^2 = 7. \]

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