Sketch the three basic conic sections in standard position with vertices and foci on the x-axis.

York 2021-08-10 Answered
Sketch the three basic conic sections in standard position with vertices and foci on the x-axis.

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Expert Answer

comentezq
Answered 2021-08-11 Author has 19928 answers

Step 1
The basic three conic sections are parabola. ellipse and hyperbola.
The standard equation of the parabola having its vertex and focus on the x-axis is
\(\displaystyle{y}^{{{2}}}={4}{a}{x}.\)
The focus of the parabola is \(\displaystyle{\left({a},\ {0}\right)}\) and the vertex is \(\displaystyle{\left({0},\ {0}\right)}.\)
Sketch the graph of the parabola \(\displaystyle{y}^{{{2}}}={4}{a}{x}\) as shown below.
image

From the above figure, it is observed that the standard parabola is symmetric about x-axis and opening right side.
Step 2
The standard equation of the ellipse having its vertices and foci on the x-axis is
\(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}.\ {a}^{{{2}}}={b}^{{{2}}}+{c}^{{{2}}}\)
The vertices of the ellipse are \(\displaystyle{\left(\pm\ {a},\ {0}\right)}\) and the foci are \(\displaystyle{\left(\pm\ {c},\ {0}\right)}.\)
Draw the ellipse \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}+{\frac{{{y}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}\) as shown below.
image From the above figure, it is observed that the ellipse has the x-axis as the major axis and the y-axis as the minor axis. And it is called the horizontal ellipse.
Step 3
The standard equation of the hyperbola having its vertices and foci on the x-axis is
\(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{y}^{{{2}}}}}{{{b}^{{{2}}}}}}={z}.\ {b}^{{{2}}}={c}^{{{2}}}-{a}^{{{2}}}.\)
The vertices of the hyperbola are \(\displaystyle{\left(\pm\ {a},\ {0}\right)}\) and the foci are \(\displaystyle{\left(\pm\ {c},\ {0}\right)}\)
Draw the hyperbola \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{y}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}\) as shown below.
image

From the above figure, it is observed that the hyperbola has the x-axis as major axis and the y-axis as a minor axis.
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