To determine: The conic section and to find the vertices and foci: x2 - y2y=4

To determine: The conic section and to find the vertices and foci: x2 - y2y=4

Question
Conic sections
asked 2020-12-24
To determine: The conic section and to find the vertices and foci: \(\displaystyle{x}{2}\ -\ {y}{2}{y}={4}\)

Answers (1)

2020-12-25
The equation \(\displaystyle{x}{2}\ -\ {2}{y}{2}={4}\) can be written as: \(\displaystyle{x}{2}\ -\ {2}{y}{2}={4}{x}{24}\ -\ {2}{y}{24}={2}{\left({x}{2}\right)}{2}\ -\ {\left({y}{2}\right)}{2}={1}\) The standard equation of hyperbola with foci \(\displaystyle{\left(\pm\ {c},\ {0}\right)}\) will be given by: \(\displaystyle{\left({x}{a}\right)}{2}\ -\ {\left({y}{b}\right)}{2}={1}\) Thus, on comparing equations \(\displaystyle{x}{2}\ -\ {2}{y}{2}={4}{x}{24}\ -\ {2}{y}{24}={2}{\left({x}{2}\right)}{2}\ -\ {\left({y}{2}\right)}{2}={1}\) and \(\displaystyle{\left({x}{a}\right)}{2}\ -\ {\left({y}{b}\right)}{2}={1}\) we find that the given conic is a hyperbola. Now, compare equation \(\displaystyle{x}{2}\ -\ {2}{y}{2}={4}{x}{24}\ -\ {2}{y}{24}={2}{\left({x}{2}\right)}{2}\ -\ {\left({y}{2}\right)}{2}={1}\) and \(\displaystyle{\left({x}{a}\right)}{2}\ -\ {\left({y}{b}\right)}{2}={1}\) and we get: \(\displaystyle{a}={2}\ \text{and}\ {b}={2}\) Now, \(\displaystyle{c}={a}{2}\ +\ {b}{2}={4}\ +\ {2}={6}\) Thus, the vertices of the given conic are \(\displaystyle{\left(\pm\ {2},\ {0}\right)}\) The foci of the given conic is \(\displaystyle{\left(\pm\ {6},\ {0}\right)}\) Conclusion: Hence, the given conic section is a hyperbola. The vertices of hyperbola are \(\displaystyle{\left(\pm\ {2},\ {0}\right)}\) and the foci of the hyperbola is \(\displaystyle{\left(\pm\ {6},\ {0}\right)}\).
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