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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.   

Show Worked Solution
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-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`