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.

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.

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.