Graphing Circles
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Question 1 of 4
1. Question
Identify the centre and radius, then sketch the following circle.`(x+2)^2+(y-4)^2=25`Hint
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In order to graph the circle, first find the centre and radiusThe centre and radius can be read from the standard form equation`(x-``h``)^2+(y-``k``)^2=``r^2`where the centre is `(``h``,``k``)` and the radius is `r`Rewrite the given equation in standard form`(x-``(-2)``)^2+(y-``4``)^2=``5^2`In this equation `h=-2``,``k=4` and `r=5`The centre is `(``-2``,``4``)` and the radius is `5`To sketch the circle first plot the centre `(``-2``,``4``)`.Since the radius is `5`, plot the points which are `5` units above, below, left and right of the centre.
Complete the sketch
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Question 2 of 4
2. Question
Find the radius of the circle which has centre `(-1,-5)` and passes through `(2,-1)`. Then sketch this circle.Hint
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In order to graph the circle, first find the radiusThe centre and radius can be read from the standard form equation`(x-``h``)^2+(y-``k``)^2=``r^2`where the centre is `(``h``,``k``)` and the radius is `r`.The centre of our circle is `(``-1``,``-5``)`, so in the standard equation `h=-1`` and ``k=-5``(x-``(-1)``)^2+(y-``-5``)^2=``r^2``(x+1)^2+(y+5)^2=``r^2`We know the point `(``2``,``-1``)`, lies on the circle, so we can substitute `x=2` and `y=-1` into the equation and solve to find `r`.`(``2``+1)^2+(``-1``+5)^2=``r^2``(3)^2+(4)^2=``r^2``9+16=``r^2``25=``r^2``r=5`To sketch the circle first plot the centre `(``-1``,``-5``)`.Since the radius is `5`, plot the points which are `5` units above, below, left and right of the centre.
Complete the sketch
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Question 3 of 4
3. Question
Identify the centre and radius, then sketch the following circle.`2x^2+4x+2y^2-16y+18=0`Hint
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In order to graph the circle, first find the centre and radiusThe centre and radius can be read from the standard form equation`(x-``h``)^2+(y-``k``)^2=``r^2`where the centre is `(``h``,``k``)` and the radius is `r`Rewrite the given equation in standard form.Start by dividing both sides of the equation by 2.`2x^2+4x+2y^2-16y+18=0``x^2+2x``+``y^2-8y``+9=0`Now complete the squares.Start by adding constants to the expressions `x^2+2x` and `y^2-8y`.The constant to add to `x^2+2x` is half the coefficient of `x` squared.`(\frac{1}{2}\times2)^2=1^2=``1`The constant to add to `y^2-8y` is half the coefficient of `y` squared.`(\frac{1}{2}\times(-8))^2=4^2=``16`Since you are adding `1` and `16` to the left-hand side, add `1``+``16``=17` to the right-hand side`x^2+2x+1``+``y^2-8x+16``+9=17`Now use the facts that `x^2+2x+1=(x+1)^2` and `y^2-8x+16=(y-4)^2` to rewrite these parts in the equation.`(x+1)^2``+``(y-4)^2``+9=17`Subtract `9` from both sides of the equation and finish rewriting in standard form.`(x+1)^2+(y-4)^2=8``(x-``(-1)``)^2+(y-``4``)^2=``\sqrt{8}^2`In this equation `h=-1``,``k=4` and `r=\sqrt{8}`The centre is `(``-1``,``4``)` and the radius is `\sqrt{8}\approx 2.828`To sketch the circle first plot the centre `(``-1``,``4``)`.Since the radius is `2.828`, plot the points which are `2.828` units above, below, left and right of the centre.
Complete the sketch
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Question 4 of 4
4. Question
Identify the centre and radius, then sketch the following circle.`x^2+y^2-4x+6y-3=0`Hint
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In order to graph the circle, first find the centre and radiusThe centre and radius can be read from the standard form equation`(x-``h``)^2+(y-``k``)^2=``r^2`where the centre is `(``h``,``k``)` and the radius is `r`Rewrite the left-hand side so the terms with the same variables are together.`x^2-4x``+``y^2+6y``-3=0`Now complete the squares.Start by adding constants to the expressions `x^2-4x` and `y^2+6y`.The constant to add to `x^2-4x` is half the coefficient of `x` squared.`(\frac{1}{2}\times (-4))^2=2^2=``4`The constant to add to `y^2+6y` is half the coefficient of `y` squared.`(\frac{1}{2}\times 6)^2=3^2=``9`Since you are adding `4` and `9` to the left-hand side, add `4``+``9``=13` to the right-hand side`x^2-4x+4``+``y^2+6y+9``-3=13`Now use the facts that `x^2-4x+4=(x-2)^2` and `y^2+6x+9=(y+3)^2` to rewrite these parts in the equation.`(x-2)^2``+``(y+3)^2``-3=13`Add `3` to both sides of the equation and finish rewriting in standard form.`(x-2)^2+(y+3)^2=16``(x-``2``)^2+(y-``(-3)``)^2=``4^2`In this equation `h=2``,``k=-3` and `r=4`The centre is `(``2``,``-3``)` and the radius is `4`.To sketch the circle first plot the centre `(``2``,``-3``)`.Since the radius is `4`, plot the points which are `4` units above, below, left and right of the centre.
Complete the sketch