What is an ellipse.

Conic sections
asked 2021-08-14
What is an ellipse.

Answers (1)

We know that a conic sections are curves that result from the intersection of a right circular cone and a plane. There are four conic sections: the circle, the ellipse, the parabola and the hyperbola.
An ellipse is the set of all points P in a plane the sum of whose distances from two fixed.
Points \(\displaystyle{F}_{{{1}}}\) and \(\displaystyle{F}_{{{2}}}\) is constant. Those two fixed points are known as foci of the ellipse. And the midpoint of two foci is the center of the ellipse.
Line-segment joining the vertices of the ellipse is known as its major axis and the line segment perpendicular to the major axis and passing through center is known as its minor axis.
The standard form of the equation of an ellipse with center at origin, major axis of length \(\displaystyle{2}{a}\) and minor axis of lenght \(\displaystyle{2}{b}\) is
\(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}\) or \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{b}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{a}^{{{2}}}}}}={1}\)
For the equation \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{a}^{{{2}}}}}}={1}\), the major axis of the ellipse will be horizontal and for the equation \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{b}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{a}^{{{2}}}}}}={1}\), the major axis of the ellipse will be vertical.
Compare the provided equation \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{16}}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}{\left\lbrace{4}\right\rbrace}={1}\) with the standard form \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{b}^{{{2}}}}}}={1},\)
\(\displaystyle{a}^{{{2}}}={16}\Rightarrow\ {a}={4}\)
\(\displaystyle{b}^{{{2}}}={4}\Rightarrow\ {b}={2}\)
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