Exercise · ZW-PARAMETRIC-CURVE-007 · ★★☆ Intermediate
Which Cartesian equation do the points of \(x=3\cos t,\ y=2\sin t\) satisfy?
Choose one option.
Show answer
- D. \(\dfrac{x^2}{9}+\dfrac{y^2}{4}=1\)
Explanation
Solving, \(\cos t=x/3\) and \(\sin t=y/2\); using \(\cos^2t+\sin^2t=1\) gives \(\frac{x^2}{9}+\frac{y^2}{4}=1\). The semi-axes are 3 along \(x\) and 2 along \(y\).
Review the concept: Parametric curve
A parametric curve is the path traced by a point in space when its coordinates depend on a parameter \(t\) through a vector-valued function \(\mathbf r\).
\[\mathbf r(t)=\langle x(t),y(t),z(t)\rangle,\quad a\le t\le b\]
More practice problems: parametric curve
- ZW-PARAMETRIC-CURVE-001 Let mathbf r(t)=langle 2t, t^2rangle. Which point corresponds to t=3?
- ZW-PARAMETRIC-CURVE-002 Eliminate the parameter in x=t+1,\ y=2t. Which Cartesian equation results?
- ZW-PARAMETRIC-CURVE-003 What curve does mathbf r(t)=langlecos t,sin trangle trace for 0≤ t≤ 2π?
- ZW-PARAMETRIC-CURVE-004 Which space curve is mathbf r(t)=langlecos t,sin t,trangle?
- ZW-PARAMETRIC-CURVE-005 If mathbf r(t)=langle t^2, t^3rangle, what is mathbf r'(1)?
- ZW-PARAMETRIC-CURVE-006 What is the tangent line to mathbf r(t)=langle t, t^2rangle at t=1?
- ZW-PARAMETRIC-CURVE-008 Which one parametrizes the segment from (1,2,0) to (3,0,4) with 0≤ t≤1?
- ZW-PARAMETRIC-CURVE-009 Let mathbf r_1(t)=langle t,t^2rangle and mathbf r_2(t)=langle t^2,t^4rangle, with t∈mathbb R. Which…