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Circles and Hyperbolas, SMB-019

Find the centre and radius of the circle with the equation

     `x^2+6x+y^2-y+3=0`  (2 marks)

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`text{Centre}\ (-3,1/2),\ text{Radius}\ = 5/2`     

Show Worked Solution
`x^2+6x+y^2-y+3` `=0`  
`x^2+6x+9+y^2-y+1/4-25/4` `=0`  
`(x+3)^2+(y-1/2)^2` `=25/4`  
`(x+3)^2+(y-1/2)^2` `=(5/2)^2`  

 
`text{Centre}\ (-3,1/2),\ text{Radius}\ = 5/2`     

Circles and Parabolas, SMB-018

  1. Find the centre and radius of the circle with the equation
  2.      `x^2-2x+y^2+3y-3/4=0`  (2 marks)

  3. Sketch the circle on the graph below.  (1 mark)
     

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a.   `text{Centre}\ (1,-3/2),\ \ text{Radius}\ =2`

b.  


        

Show Worked Solution
a.    `x^2-2x+y^2+3y-3/4` `=0`
  `x^2-2x+1+y^2+3y+9/4-4` `=0`
  `(x-1)^2+(y+3/2)^2` `=2^2`

 
`text{Centre}\ (1,-3/2),\ \ text{Radius}\ =2`

b.   

 

Circles and Hyperbolas, SMB-017

  1. Find the centre and radius of the circle with the equation
  2.      `x^2-4x+y^2+6y+9=0`  (2 marks)

  3. Sketch the circle on the graph below.  (1 mark)

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a.   `text{Centre}\ (2,-3),\ text{Radius}\ = 2`

b.   


        

Show Worked Solution
a.    `x^2-4x+y^2+6y+9` `=0`
  `x^2-4x+4+y^2+6y+9-4` `=0`
  `(x-2)^2+(y+3)^2` `=2^2`

  
`:.\ text{Centre}\ (2,-3),\ text{Radius}\ = 2`

b.   

Circles and Hyperbolas, SMB-014

  1. Find the centre and radius of the circle with the equation
  2.      `x^2+6x+y^2+4y+4=0`  (2 marks)

  3. Sketch the circle on the graph below.  (1 mark)
     

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a.   `(x+3)^2 + (y+2)^2 = 3^2`

`:.\ text(Centre:)\ (-3,-2)\text(, Radius:)\ 3`

b.   

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a.    `x^2+6x+y^2+4y+4` `=0`
  `x^2+6x+9+y^2+4y+4-9` `=0`
  `(x+3)^2+(y+2)^2` `=3^2`

  
`:.\ text(Centre:)\ (-3,-2)\text(, Radius:)\ 3`

  
b.   

Circles and Hyperbolas, SMB-013

  1. Write down the equation of the circle with centre `(1, -2)` and radius 2.  (1 mark)

  2. On the graph, sketch the circle in part (a).  (2 marks)

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a.   `(x-1)^2 + (y+2)^2 = 4`

b.    
     

Show Worked Solution

a.   `text{Circle with centre}\ (1,-2),\ r = 2:`

`(x-1)^2 + (y+2)^2 = 4`
 

b.   

Circles and Hyperbolas, SMB-010 MC

A circle with centre `(a,-2)` and radius 5 units has equation

`x^2-6x + y^2 + 4y = b`  where  `a`  and  `b`  are real constants.

The values of  `a`  and  `b`  are respectively

  1. −3 and 38
  2. 3 and 12
  3. −3 and −8
  4. 3 and 18
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`B`

Show Worked Solution

`x^2-6x + y^2 + 4y=b`

`text(Completing the squares:)`

`x^2-6x + 3^2-9 + y^2 + 4y + 2^2-4` `= b`
`(x-3)^2 + (y + 2)^2-13` `= b`
`(x-3)^2 + (y + 2)^2` `= b + 13`

 
`:. a=3`

`:. b+13=25\ \ =>\ \ b=12`

`=> B`

Circles and Hyperbolas, SMB-009

Sketch the graph of  `y=4/(x-3)`.  (3 marks)

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Show Worked Solution

`text{Vertical asymptote at}\ \ x=3`

`text{As}\ \ x->oo, \ y->0`

`text{Horizontal asymptote at}\ \ y=0`

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 & \ \ 0\ \  & \ \ 2\ \  & \ \ 4\ \  & \ \ 5\ \  \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -1 & -\frac{4}{3} & -4 & 4 & 2\\
\hline
\end{array}

 

Circles and Hyperbolas, SMB-008

Sketch the graph of  `y=2/(3-x)`.  (3 marks)
 

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Show Worked Solution

`text{Vertical asymptote at}\ \ x=3`

`text{As}\ \ x->oo, \ y->0`

`text{Horizontal asymptote at}\ \ y=0`

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 & \ \ 0\ \  & \ \ 2\ \  & \ \ 4\ \  & \ \ 5\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & \frac{1}{2} & \frac{2}{3} & 2 & -2 & -1\\
\hline
\end{array}

 

Circles and Hyperbolas, SMB-007

Sketch the graph of  `y=3/(x+1)`.  (2 marks)
 

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Show Worked Solution

`text{Vertical asymptote at}\ \ x=-1`

`text{As}\ \ x->oo, \ y->0`

`text{Horizontal asymptote at}\ \ y=0`

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -3  & -2  & -1  & \ \ 0\ \  & \ \ 1\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -\frac{3}{2} & -3 & ∞ & 3 & \frac{3}{2} \\
\hline
\end{array}

Circles and Hyperbolas, SMB-006

Sketch the graph of  `y=1/(x-2)`.  (2 marks)

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Show Worked Solution

`text{Vertical asymptote at}\ \ x=2`

`text{As}\ \ x->oo, \ y->0`

`text{Horizontal asymptote at}\ \ y=0`

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \  & \ \ 3\ \  & \ \ 4\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -\frac{1}{2} & -1 & ∞ & 1 & \frac{1}{2} \\
\hline
\end{array}

Circles and Hyperbolas, SMB-005

Sketch the graph of  `f(x) = (2x+1)/(x-1)`. Label the axis intercepts with their coordinates and label any asymptotes with the appropriate equation.  (4 marks)

  

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Show Worked Solution
`(2x+1)/(x-1)` `=(2x-2+3)/(x-1)`  
  `=(2(x-1)+3)/(x-1)`  
  `=2 + 3/(x-1)`  

COMMENT: Manipulation of the equation (as shown) makes graphing much easier.

 
`text(Asymptotes:)\ \ x = 1,\ \ y = 2`

`text(As)\ \ x->oo,\ \ y->2(+)`

`text(As)\ \ x->-oo,\ \ y->2(-)`

`text(As)\ \ x->-1 (-),\ \ y->-oo`

`text(As)\ \ x->-1 (+),\ \ y->oo`

Circles and Hyperbolas, SMB-004

  1. List all asymptotes of the graph  `y=2-1/x`.  (2 marks)

  2. Hence, sketch the graph of  `y=2-1/x`.  (2 marks)

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i.    `text{Vertical asymptote at}\ \ x=0`

`text{Horizontal asymptote at}\ \ y=2`

ii.

Show Worked Solution

i.     `y=2-1/x`

`text{Vertical asymptote at}\ \ x=0`

`text{As}\ x->oo, \ 1/x -> 0\ \ => 2-1/x -> 2`

`text{Horizontal asymptote at}\ \ y=2`

ii.  

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & \frac{5}{2} & 3 & ∞ & 1 & \frac{3}{2} \\
\hline
\end{array}

Circles and Hyperbolas, SMB-003

Sketch the graph of  `y=2/x+2`.

Clearly mark all asymptotes.  (3 marks)

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Show Worked Solution

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 1 & 0 & ∞ & 4 & 3 \\
\hline
\end{array}

Circles and Hyperbola, SMB-002

Sketch the graph of  `y=-2/x`.  (2 marks)

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Show Worked Solution

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 1 & 2 & ∞ & -2 & -1 \\
\hline
\end{array}

Circles and Hyperbola, SMB-001

Sketch the graph of  `y=3/x`.  (2 marks)

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Show Worked Solution

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 & \ \ 0\ \  & \ \ 1\ \  & \ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex}\ \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -\frac{3}{2} & -3 & ∞ & 3 & \frac{3}{2} \\
\hline
\end{array}

Algebra, STD2 A4 2022 HSC 24

A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating `M` and `T`.   (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from `T=5^@C` to `T=30^@ text{C}.`   (2 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \  & \ \ 15\ \ \  & \ \ \ 30\ \ \  \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}

 
           

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a.  `M prop 1/T \ \ =>\ \ M=k/T`

  `12` `=k/15`
  `k` `=15 xx 12=180`

 
`:.M=180/T`

b.   

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \  & \ \ 15\ \ \  & \ \ 30\ \  \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}

 

     

Show Worked Solution
a.    `M` `prop 1/T`
  `M` `=k/T`
  `12` `=k/15`
  `k` `=15 xx 12`
    `=180`

 
`:.M=180/T`


♦♦ Mean mark part (a) 29%.

b.   

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \  & \ \ 15\ \ \  & \ \ 30\ \  \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}

 

     


♦ Mean mark 44%.

Functions, 2ADV F2 2021 HSC 19

Without using calculus, sketch the graph of  `y = 2 + 1/(x + 4)`, showing the asymptotes and the `x` and `y` intercepts.  (3 marks)

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Show Worked Solution

`text(Asymptotes:)\ x = -4`

`text(As)\ \ x -> ∞, y -> 2`

`ytext(-intercept occurs when)\ \ x = 0:`

`y = 2.25`

`xtext(-intercept occurs when)\ \ y = 0:`

`2 + 1/(x + 4) = 0 \ => \ x = -4.5`
 

Functions, 2ADV F1 2020 HSC 24

The circle of  `x^2-6x + y^2 + 4y-3 = 0`  is reflected in the `x`-axis.

Sketch the reflected circle, showing the coordinates of the centre and the radius.  (3 marks)

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Show Worked Solution
`x^2-6x + y^2 + 4y-3` `= 0`
`x^2-6x + 9 + y^2 + 4y + 4-16` `= 0`
`(x-3)^2 + (y + 2)^2` `= 16`

 
`=>\  text{Original circle has centre (3, − 2), radius = 4}`

`text(Reflect in)\ xtext(-axis):`

♦ Mean mark 48%.

`text{Centre (3, − 2) → (3, 2)}`
 

Functions, 2ADV F1 2016 HSC 11a

Sketch the graph of  `(x-3)^2 + (y + 2)^2 = 4.`  (2 marks)

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Show Worked Solution

`(x-3)^2 + (y + 2)^2 = 4\ \ text(is a circle),`

`text(centre)\ (3, -2),\ text(radius 2.)`
 

Algebra, STD2 A4 2014 HSC 29a

The cost of hiring an open space for a music festival is  $120 000. The cost will be shared equally by the people attending the festival, so that  `C`  (in dollars) is the cost per person when  `n`  people attend the festival.

  1. Complete the table below by filling in the THREE missing values.   (1 mark)
    \begin{array} {|l|c|c|c|c|c|c|}
    \hline
    \rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
    \hline
    \rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} &  &  &  & 60 & 48\ & 40 \ \\
    \hline
    \end{array}
  2. Using the values from the table, draw the graph showing the relationship between  `n`  and  `C`.   (2 marks)
     
  3. What equation represents the relationship between `n` and `C`?   (1 mark)

  4. Give ONE limitation of this equation in relation to this context.   (1 mark)

  5. Is it possible for the cost per person to be $94? Support your answer with appropriate calculations.   (1 mark)

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i.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

ii. 

iii.   `C = (120\ 000)/n`

`n\ text(must be a whole number)`
 

iv.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`
 

v.   `text(If)\ C = 94:`

`94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`

 
`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`

Show Worked Solution

i.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

ii. 

 

♦ Mean mark (iii) 48%

iii.   `C = (120\ 000)/n`

 

♦♦♦ Mean mark (iv) 7%
COMMENT: When asked for limitations of an equation, look carefully at potential restrictions with respect to both the domain and range.

iv.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`

 

v.   `text(If)\ C = 94`

`=> 94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`
♦ Mean mark (v) 38%

 

`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`

Functions, 2ADV F1 2010 HSC 1c

Write down the equation of the circle with centre `(-1, 2)` and radius 5.   (1 mark)

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 `text{Circle with centre}\  (-1,2),\ r = 5`

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

Show Worked Solution
 MARKER’S COMMENT: Expanding this equation is not necessary!

`text{Circle with centre}\ (-1, 2),\ r = 5`

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