The ellipse and hyperbola Quiz

1. The Cartesian equation of $x=2\cos t$, $y=5\sin t$ is

• A. $\frac{x^2}{25}+\frac{y^2}{4}=1$
• B. $\frac{x^2}{4}+\frac{y^2}{25}=1$
• C. $\frac{x^2}{2}+\frac{y^2}{5}=1$
• D. $2x^2+5y^2=1$

2. Which parametrisation traces only the right branch of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$?

• A. $x=a\cos t$, $y=b\sin t$
• B. $x=a\sec t$, $y=b\tan t$ for all $t$
• C. $x=a\sinh t$, $y=b\cosh t$
• D. $x=a\cosh t$, $y=b\sinh t$

3. To derive the Cartesian equation from $x=a\sec t$, $y=b\tan t$ you use

• A. $1+\tan^2t=\sec^2t$
• B. $\sin^2t+\cos^2t=1$
• C. $\cosh^2t-\sinh^2t=1$
• D. $\cos2t=1-2\sin^2t$

4. For the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, the semi-major axis has length

• A. $2$
• B. $4$
• C. $3$
• D. $9$

5. Which parametrisation gives the left branch of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$?

• A. $x=a\cosh t$, $y=-b\sinh t$
• B. $x=-a\cosh t$, $y=b\sinh t$
• C. $x=a\sinh t$, $y=b\cosh t$
• D. $x=a\operatorname{sech}t$, $y=b\tanh t$