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Functions, EXT1 F1 EQ-Bank 4 MC

Determine the Cartesian equation of the circle given by the parametric equations

\(\begin{aligned} & x=-3+4 \cos \theta \\
& y=1+4 \sin \theta\end{aligned}\)

  1. \((x+3)^2+(y-1)^2=4\)
  2. \((x-3)^2+(y+1)^2=4\)
  3. \((x+3)^2+(y-1)^2=16\)
  4. \((x-3)^2+(y+1)^2=16\)
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\(C\)

Show Worked Solution

\(x=-3+4 \cos\, \theta\ \Rightarrow \ \cos\,\theta=\dfrac{x+3}{4} \)

\(y=1+4 \sin\, \theta\ \Rightarrow \ \sin\,\theta=\dfrac{y-1}{4} \)

\(\text{Using}\ \ \sin^{2}\theta + \cos^{2}\theta=1:\)

\( \Big(\dfrac{x+3}{4}\Big)^{2} + \Big(\dfrac{y-1}{4}\Big)^{2}\) \(=1\)  
\((x+3)^2+(y-1)^2\) \(=16\)  

 
\(\Rightarrow C\)

Functions, EXT1 F1 2021 SPEC2 7

A function is defined parametrically by

   `x(t) = 5cos(2t) + 1,\ \ y(t) = 5sin(2t)-1`

If  `A(6, –1)`  and  `B(1, 4)`  are two points that lie on the graph of the function, then find the shortest distance along the graph from `A` to `B`.   (2 marks)

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`(5pi)/2`

Show Worked Solution

`x = 5cos(2t) + 1 \ => \ cos(2t) = (x-1)/5`

`y = 5sin(2t)-1 \ => \ sin(2t) = (y + 1)/5`

`(cos(2t))^2 + (sin(2t))^2` `=1`  
`(x-1)^2 + (y + 1)^2` `=25`  

 

`text(Distance)` `= 1/4 xx 2 xx pi xx r`
  `= (5pi)/2`

Functions, EXT1 F1 2021 HSC 8 MC

The diagram shows a semicircle.
 

Which pair of parametric equations represents the semicircle shown?

  1. `{(x = 3 + sin t),(y = 2 + cos t):} \ \ \ \ \ text(for)\ \ -pi/2 <= t <= pi/2`
  2. `{(x = 3 + cos t),(y = 2 + sin t):} \ \ \ \ \ text(for)\ \ -pi/2 <= t <= pi/2`
  3. `{(x = 3 - sin t),(y = 2 - cos t):} \ \ \ \ \ text(for)\ \ -pi/2 <= t <= pi/2`
  4. `{(x = 3 - cos t),(y = 2 - sin t):} \ \ \ \ \ text(for)\ \ -pi/2 <= t <= pi/2`
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`C`

Show Worked Solution

`text(By elimination:)`

♦♦ Mean mark 31%.

`text(When)\ \ t = pi/2,`

`A(4, 2), \ B(3, 3), \ C(2, 2), \ D(3, 1)`

`->\ text(Eliminate B)`
 

`text(When)\ \ t = -pi/2,`

`A(2, 2), \ C(4, 2), \ D(3, 3)`

`->\ text(Eliminate D)`
 

`text(When)\ \ t = 0,`

`A(3, 3), \ C(3, 1)`

`->\ text(Eliminate A)`
 

`=>\ C`

Functions, EXT1 F1 EQ-Bank 10

An equation can be expressed in the parametric form

`x = 2costheta - 1`

`y = 2 + 2sintheta`

  1.  Express the equation in Cartesian form.  (2 marks)

  2.  Sketch the graph.  (1 mark)

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  1. `(x + 1)^2 + (y – 2)^2 = 4`
  2.    
Show Worked Solution
i.    `2costheta` `= x + 1`
  `costheta` `= (x + 1)/2`
`2sintheta` `= y – 2`
`sintheta` `= (y – 2)/2`

 
`text(Using)\ \ cos^2theta + sin^2 = 1:`

`((x + 1)/2)^2 + ((y – 2)/2)^2` `= 1`
`(x + 1)^2 + (y – 2)^2` `= 4`

 

ii.  `text{Sketch circle with centre (−1, 2),  radius = 2}`
 

Functions, EXT1 F1 SM-Bank 8

A circle has the equation  `x^2 - 10x + y^2 + 6y +25 = 0`

  1.  Express the circle in parametric form.  (2 marks)

  2.  Sketch the circle.  (1 mark)

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  1. `x = 5 + 3costheta`
    `y = −3 + 3sintheta`
  2.   
Show Worked Solution
i.    `x^2 – 10x + y^2 + 6y+25` `= 0`
  `(x – 5)^2 + (y + 3)^2 – 9` `= 0`
  `(x – 5)^2 + (y + 3)^2` `= 9`

 
`=>\ text{Circle centre (5, −3),  radius 3}`
 

`:.\ text(Parametric form is:)`

`x = 5 + 3costheta`

`y = −3 + 3sintheta`

 

ii.  

Functions, EXT1 F1 SM-Bank 4

Sketch the curve described by these two parametric equations

`x = 3cost + 2`

`y = 3sint - 3`   for   `0 <= t < 2pi`.  (3 marks)

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Show Worked Solution

`(x – 2) = 3cost`

`(y + 3) = 3sint`

`(x – 2)^2 + (y + 3)^2` `= (3cost)^2 + (3sint)^2`
  `= 9cos^2t + 9sin^2t`
  `= 9(cos^2t + sin^2t)`
  `= 9`

 
`text(Sketch:) \ (x – 2)^2 + (y + 3)^2 = 3^2`