Question

question 6
the graph below can be represented by parametric equations of the form
\

$$\begin{cases}x(t) = a\\cos(t)\\\\y(t) = b\\sin(t)\\end{cases}$$

if a and b are positive values, a = \square and b = \square
question help: video

Explanation:

Step1: Determine the value of \(a\)

For the parametric equation \(x(t) = a\cos(t)\), the maximum value of \(x(t)\) occurs when \(\cos(t)=1\), so \(x = a\). From the graph, the rightmost point of the ellipse (which is the maximum \(x\)-value) is at \(x = 4\). So \(a = 4\).

Step2: Determine the value of \(b\)

For the parametric equation \(y(t)=b\sin(t)\), the maximum value of \(y(t)\) occurs when \(\sin(t) = 1\), so \(y = b\). From the graph, the topmost point of the ellipse (which is the maximum \(y\)-value) is at \(y = 2\). So \(b = 2\).

Answer:

\(a = 4\) and \(b = 2\)