Derive the Equation of a Circle - Geometry
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What is the standard form after completing the square for $x^2+y^2+8x+2y-8=0$?
What is the standard form after completing the square for $x^2+y^2+8x+2y-8=0$?
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$(x+4)^2+(y+1)^2=25$. Complete square: $(x+4)^2-16+(y+1)^2-1-8=0$.
$(x+4)^2+(y+1)^2=25$. Complete square: $(x+4)^2-16+(y+1)^2-1-8=0$.
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Identify the geometric figure given by $(x-1)^2+(y+9)^2=0$.
Identify the geometric figure given by $(x-1)^2+(y+9)^2=0$.
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A single point at $(1,-9)$. Circle with radius $0$ is just the center point.
A single point at $(1,-9)$. Circle with radius $0$ is just the center point.
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What is the center and radius of the circle $(x+2)^2+(y-1)^2=16$?
What is the center and radius of the circle $(x+2)^2+(y-1)^2=16$?
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Center $(-2,1)$, radius $4$. From $(x+2)^2=(x-(-2))^2$ and $r=\sqrt{16}=4$.
Center $(-2,1)$, radius $4$. From $(x+2)^2=(x-(-2))^2$ and $r=\sqrt{16}=4$.
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What is the standard form equation of a circle with center $(h,k)$ and radius $r$?
What is the standard form equation of a circle with center $(h,k)$ and radius $r$?
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$(x-h)^2+(y-k)^2=r^2$. Standard form derived from the distance formula.
$(x-h)^2+(y-k)^2=r^2$. Standard form derived from the distance formula.
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What is the standard form of $x^2+y^2+10x-2y+1=0$?
What is the standard form of $x^2+y^2+10x-2y+1=0$?
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$(x+5)^2+(y-1)^2=25$. From general form after completing the square.
$(x+5)^2+(y-1)^2=25$. From general form after completing the square.
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What is the center and radius of $x^2+y^2+10x-2y+1=0$?
What is the center and radius of $x^2+y^2+10x-2y+1=0$?
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Center $(-5,1)$, radius $5$. Complete square: $(x+5)^2-25+(y-1)^2-1+1=0$.
Center $(-5,1)$, radius $5$. Complete square: $(x+5)^2-25+(y-1)^2-1+1=0$.
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What is the center and radius of the circle $ (x-3)^2 + (y+5)^2 = 49 $?
What is the center and radius of the circle $ (x-3)^2 + (y+5)^2 = 49 $?
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Center $ (3, -5) $, radius $ 7 $. Read center from $ (x-h)^2 + (y-k)^2 = r^2 $ and find $ r = \sqrt{49} = 7 $.
Center $ (3, -5) $, radius $ 7 $. Read center from $ (x-h)^2 + (y-k)^2 = r^2 $ and find $ r = \sqrt{49} = 7 $.
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What is the equation of the circle with center $ (0,-3) $ and radius $ \sqrt{10} $?
What is the equation of the circle with center $ (0,-3) $ and radius $ \sqrt{10} $?
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$x^2+(y+3)^2=10$. Center $ (0,-3) $ gives $x^2+(y+3)^2=(\sqrt{10})^2$.
$x^2+(y+3)^2=10$. Center $ (0,-3) $ gives $x^2+(y+3)^2=(\sqrt{10})^2$.
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What number is added to complete the square for $x^2+10x$?
What number is added to complete the square for $x^2+10x$?
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Add $25$. Take half of coefficient: $ (\frac{10}{2})^2 = 25$.
Add $25$. Take half of coefficient: $ (\frac{10}{2})^2 = 25$.
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What are the center and radius of $x^2+y^2-6x+8y-11=0$?
What are the center and radius of $x^2+y^2-6x+8y-11=0$?
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Center $(3,-4)$, radius $6$. From completed square form $(x-3)^2+(y+4)^2=36$.
Center $(3,-4)$, radius $6$. From completed square form $(x-3)^2+(y+4)^2=36$.
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What is the equation of a circle with center $(4,-1)$ and radius $6$?
What is the equation of a circle with center $(4,-1)$ and radius $6$?
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$(x-4)^2+(y+1)^2=36$. Substitute $(h,k)=(4,-1)$ and $r^2=36$ into standard form.
$(x-4)^2+(y+1)^2=36$. Substitute $(h,k)=(4,-1)$ and $r^2=36$ into standard form.
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What is the equation of a circle with center $(-3,2)$ and radius $5$?
What is the equation of a circle with center $(-3,2)$ and radius $5$?
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$(x+3)^2+(y-2)^2=25$. Use $(h,k)=(-3,2)$ and $r^2=25$ in standard form.
$(x+3)^2+(y-2)^2=25$. Use $(h,k)=(-3,2)$ and $r^2=25$ in standard form.
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What is the distance formula between $(x,y)$ and $(h,k)$ used to derive a circle equation?
What is the distance formula between $(x,y)$ and $(h,k)$ used to derive a circle equation?
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$d=\sqrt{(x-h)^2+(y-k)^2}$. Square root of sum of squared differences in coordinates.
$d=\sqrt{(x-h)^2+(y-k)^2}$. Square root of sum of squared differences in coordinates.
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What equation results from setting the distance from $(x,y)$ to $(h,k)$ equal to $r$ and squaring?
What equation results from setting the distance from $(x,y)$ to $(h,k)$ equal to $r$ and squaring?
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$(x-h)^2+(y-k)^2=r^2$. Setting distance equal to $r$ and squaring both sides.
$(x-h)^2+(y-k)^2=r^2$. Setting distance equal to $r$ and squaring both sides.
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What is the center and radius of $x^2+y^2=81$?
What is the center and radius of $x^2+y^2=81$?
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Center $(0,0)$, radius $9$. Circle centered at origin with $r=\sqrt{81}=9$.
Center $(0,0)$, radius $9$. Circle centered at origin with $r=\sqrt{81}=9$.
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What is the center and radius of $x^2+y^2=\frac{1}{4}$?
What is the center and radius of $x^2+y^2=\frac{1}{4}$?
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Center $(0,0)$, radius $\frac{1}{2}$. Origin center with $r=\sqrt{\frac{1}{4}}=\frac{1}{2}$.
Center $(0,0)$, radius $\frac{1}{2}$. Origin center with $r=\sqrt{\frac{1}{4}}=\frac{1}{2}$.
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What is the expanded general form of $(x-h)^2+(y-k)^2=r^2$ called?
What is the expanded general form of $(x-h)^2+(y-k)^2=r^2$ called?
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General form $x^2+y^2+Dx+Ey+F=0$. Expanded form with linear terms and constant.
General form $x^2+y^2+Dx+Ey+F=0$. Expanded form with linear terms and constant.
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What is the standard form after completing the square for $x^2+y^2-6x+8y-11=0$?
What is the standard form after completing the square for $x^2+y^2-6x+8y-11=0$?
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$(x-3)^2+(y+4)^2=36$. Complete square for $x$ and $y$ terms separately.
$(x-3)^2+(y+4)^2=36$. Complete square for $x$ and $y$ terms separately.
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What theorem justifies $ (x-h)^2+(y-k)^2=r^2 $ as the circle equation in the coordinate plane?
What theorem justifies $ (x-h)^2+(y-k)^2=r^2 $ as the circle equation in the coordinate plane?
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The Pythagorean Theorem. Distance formula is based on the Pythagorean Theorem.
The Pythagorean Theorem. Distance formula is based on the Pythagorean Theorem.
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What is the standard form after completing the square for $x^2+y^2+4x-10y+13=0$?
What is the standard form after completing the square for $x^2+y^2+4x-10y+13=0$?
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$(x+2)^2+(y-5)^2=16$. Complete square: $(x+2)^2-4+(y-5)^2-25+13=0$.
$(x+2)^2+(y-5)^2=16$. Complete square: $(x+2)^2-4+(y-5)^2-25+13=0$.
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What are the center and radius of $x^2+y^2+4x-10y+13=0$?
What are the center and radius of $x^2+y^2+4x-10y+13=0$?
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Center $(-2,5)$, radius $4$. From $(x+2)^2+(y-5)^2=16$, center $(-2,5)$, $r=4$.
Center $(-2,5)$, radius $4$. From $(x+2)^2+(y-5)^2=16$, center $(-2,5)$, $r=4$.
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What is the standard form after completing the square for $x^2+y^2-2x-12y+20=0$?
What is the standard form after completing the square for $x^2+y^2-2x-12y+20=0$?
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$(x-1)^2+(y-6)^2=17$. Complete square: $(x-1)^2-1+(y-6)^2-36+20=0$.
$(x-1)^2+(y-6)^2=17$. Complete square: $(x-1)^2-1+(y-6)^2-36+20=0$.
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What are the center and radius of $x^2+y^2-2x-12y+20=0$?
What are the center and radius of $x^2+y^2-2x-12y+20=0$?
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Center $(1,6)$, radius $\sqrt{17}$. From standard form, $r=\sqrt{17}$ (cannot simplify).
Center $(1,6)$, radius $\sqrt{17}$. From standard form, $r=\sqrt{17}$ (cannot simplify).
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What is the standard form after completing the square for $x^2+y^2+8x+2y-8=0$?
What is the standard form after completing the square for $x^2+y^2+8x+2y-8=0$?
Tap to reveal answer
$(x+4)^2+(y+1)^2=25$. Complete square: $(x+4)^2-16+(y+1)^2-1-8=0$.
$(x+4)^2+(y+1)^2=25$. Complete square: $(x+4)^2-16+(y+1)^2-1-8=0$.
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What is the standard form after completing the square for $x^2+y^2-10x-4y+13=0$?
What is the standard form after completing the square for $x^2+y^2-10x-4y+13=0$?
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$(x-5)^2+(y-2)^2=16$. Complete square: $(x-5)^2-25+(y-2)^2-4+13=0$.
$(x-5)^2+(y-2)^2=16$. Complete square: $(x-5)^2-25+(y-2)^2-4+13=0$.
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What are the center and radius of $x^2+y^2-10x-4y+13=0$?
What are the center and radius of $x^2+y^2-10x-4y+13=0$?
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Center $(5,2)$, radius $4$. From $(x-5)^2+(y-2)^2=16$, $r=\sqrt{16}=4$.
Center $(5,2)$, radius $4$. From $(x-5)^2+(y-2)^2=16$, $r=\sqrt{16}=4$.
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What is the standard form after completing the square for $x^2+y^2+6x-2y-6=0$?
What is the standard form after completing the square for $x^2+y^2+6x-2y-6=0$?
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$(x+3)^2+(y-1)^2=16$. Complete square: $(x+3)^2-9+(y-1)^2-1-6=0$.
$(x+3)^2+(y-1)^2=16$. Complete square: $(x+3)^2-9+(y-1)^2-1-6=0$.
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What are the center and radius of $x^2+y^2+6x-2y-6=0$?
What are the center and radius of $x^2+y^2+6x-2y-6=0$?
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Center $(-3,1)$, radius $4$. From $(x+3)^2+(y-1)^2=16$, $r=4$.
Center $(-3,1)$, radius $4$. From $(x+3)^2+(y-1)^2=16$, $r=4$.
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What is the standard form after completing the square for $x^2+y^2-4x+14y+29=0$?
What is the standard form after completing the square for $x^2+y^2-4x+14y+29=0$?
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$(x-2)^2+(y+7)^2=24$. Complete square: $(x-2)^2-4+(y+7)^2-49+29=0$.
$(x-2)^2+(y+7)^2=24$. Complete square: $(x-2)^2-4+(y+7)^2-49+29=0$.
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What are the center and radius of $x^2+y^2-4x+14y+29=0$?
What are the center and radius of $x^2+y^2-4x+14y+29=0$?
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Center $(2,-7)$, radius $\sqrt{24}$. From $(x-2)^2+(y+7)^2=24$, $r=\sqrt{24}$.
Center $(2,-7)$, radius $\sqrt{24}$. From $(x-2)^2+(y+7)^2=24$, $r=\sqrt{24}$.
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