Question

Equation \[x=a\cos\theta,y=b\sin\theta(a>b)\] represents a conic section whose eccentricity e is given by______.

Options

• \[\mathrm{e}^2=\frac{\mathrm{a}^2+\mathrm{b}^2}{\mathrm{a}^2}\]

• \[\mathrm{e}^2=\frac{\mathrm{a}^2+\mathrm{b}^2}{\mathrm{b}^2}\]

• \[\mathrm{e}^2=\frac{\mathrm{a}^2-\mathrm{b}^2}{\mathrm{a}^2}\]

• \[\mathrm{e}^2=\frac{\mathrm{a}^2-\mathrm{b}^2}{\mathrm{b}^2}\]

Answer 1

Equation \[x=a\cos\theta,y=b\sin\theta(a>b)\] represents a conic section whose eccentricity e is given by \[\mathrm{e}^2=\frac{\mathrm{a}^2-\mathrm{b}^2}{\mathrm{a}^2}\].

Explanation:

The parametric equations \[x=a\cos\theta,y=b\sin\theta\] with a > b represent an ellipse with:

• Major axis along the x-axis

• \( a \) = semi-major axis, \( b \) = semi-minor axis

For this ellipse, the relationship between \(a\), \(b\), and \(c\) is:

                                                                \[c^2=a^2-b^2\]

Eccentricity is defined as:

                                                              \[e=\frac{c}{a}\]

So:

                                                     \[e^2=\frac{c^2}{a^2}=\frac{a^2-b^2}{a^2}\]