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

Question
Conic sections
To determine: The conic section and to find the vertices and foci: $$\displaystyle{x}{2}\ -\ {y}{2}{y}={4}$$

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 $$\displaystyle{\left(\pm\ {6},\ {0}\right)}$$.

Relevant Questions

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