Derive the Equation of a Circle - Algebra 2
Card 1 of 30
What midpoint formula gives the center of a circle from diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
What midpoint formula gives the center of a circle from diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
Tap to reveal answer
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Midpoint formula gives center from endpoints.
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Midpoint formula gives center from endpoints.
← Didn't Know|Knew It →
What is the radius in terms of diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
What is the radius in terms of diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
Tap to reveal answer
$r=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Radius is half the distance between endpoints.
$r=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Radius is half the distance between endpoints.
← Didn't Know|Knew It →
What is the general form after expanding $x^2+(y-4)^2=20$?
What is the general form after expanding $x^2+(y-4)^2=20$?
Tap to reveal answer
$x^2+y^2-8y-4=0$. Expand and rearrange to general form.
$x^2+y^2-8y-4=0$. Expand and rearrange to general form.
← Didn't Know|Knew It →
Identify the value added to complete the square for $x^2-14x$.
Identify the value added to complete the square for $x^2-14x$.
Tap to reveal answer
$49$. Take half of coefficient: $(-14/2)^2=49$.
$49$. Take half of coefficient: $(-14/2)^2=49$.
← Didn't Know|Knew It →
What is the completed-square expression equivalent to $y^2+8y$?
What is the completed-square expression equivalent to $y^2+8y$?
Tap to reveal answer
$(y+4)^2-16$. Complete square: $(y+4)^2=y^2+8y+16$.
$(y+4)^2-16$. Complete square: $(y+4)^2=y^2+8y+16$.
← Didn't Know|Knew It →
What is the center and radius of $(x+1)^2+(y-2)^2=0$?
What is the center and radius of $(x+1)^2+(y-2)^2=0$?
Tap to reveal answer
Center $(-1,2)$; radius $0$. Degenerate circle: single point at $(-1,2)$.
Center $(-1,2)$; radius $0$. Degenerate circle: single point at $(-1,2)$.
← Didn't Know|Knew It →
What is the center and radius of $(x-7)^2+(y+3)^2=81$?
What is the center and radius of $(x-7)^2+(y+3)^2=81$?
Tap to reveal answer
Center $(7,-3)$; radius $9$. Read center and radius from standard form.
Center $(7,-3)$; radius $9$. Read center and radius from standard form.
← Didn't Know|Knew It →
What is the center and radius of $x^2+y^2-12x-2y+37=0$?
What is the center and radius of $x^2+y^2-12x-2y+37=0$?
Tap to reveal answer
Center $(6,1)$; radius $0$. Complete square: $r^2=36+1-37=0$.
Center $(6,1)$; radius $0$. Complete square: $r^2=36+1-37=0$.
← Didn't Know|Knew It →
What is the center of the circle $x^2+y^2-2x+14y+33=0$?
What is the center of the circle $x^2+y^2-2x+14y+33=0$?
Tap to reveal answer
$(1,-7)$. From standard form, center is $(1,-7)$.
$(1,-7)$. From standard form, center is $(1,-7)$.
← Didn't Know|Knew It →
What is the general form of a circle equation after expanding standard form?
What is the general form of a circle equation after expanding standard form?
Tap to reveal answer
$x^2+y^2+Dx+Ey+F=0$. Result of expanding standard form and collecting terms.
$x^2+y^2+Dx+Ey+F=0$. Result of expanding standard form and collecting terms.
← Didn't Know|Knew It →
What is the general form after expanding $(x+5)^2+(y-1)^2=36$?
What is the general form after expanding $(x+5)^2+(y-1)^2=36$?
Tap to reveal answer
$x^2+y^2+10x-2y-10=0$. Expand and rearrange to general form.
$x^2+y^2+10x-2y-10=0$. Expand and rearrange to general form.
← Didn't Know|Knew It →
What is the equation of the circle with diameter endpoints $(0,0)$ and $(6,8)$?
What is the equation of the circle with diameter endpoints $(0,0)$ and $(6,8)$?
Tap to reveal answer
$(x-3)^2+(y-4)^2=25$. Midpoint $(3,4)$, radius half diagonal length.
$(x-3)^2+(y-4)^2=25$. Midpoint $(3,4)$, radius half diagonal length.
← Didn't Know|Knew It →
What is the center of the circle with diameter endpoints $(2,-1)$ and $(8,5)$?
What is the center of the circle with diameter endpoints $(2,-1)$ and $(8,5)$?
Tap to reveal answer
$(5,2)$. Midpoint formula for diameter endpoints.
$(5,2)$. Midpoint formula for diameter endpoints.
← Didn't Know|Knew It →
What is the radius of the circle with diameter endpoints $(2,-1)$ and $(8,5)$?
What is the radius of the circle with diameter endpoints $(2,-1)$ and $(8,5)$?
Tap to reveal answer
$3\sqrt{2}$. Half the distance between diameter endpoints.
$3\sqrt{2}$. Half the distance between diameter endpoints.
← Didn't Know|Knew It →
What is the equation of the circle with diameter endpoints $(-2,4)$ and $(4,-2)$?
What is the equation of the circle with diameter endpoints $(-2,4)$ and $(4,-2)$?
Tap to reveal answer
$(x-1)^2+(y-1)^2=18$. Midpoint $(1,1)$, radius half diagonal length.
$(x-1)^2+(y-1)^2=18$. Midpoint $(1,1)$, radius half diagonal length.
← Didn't Know|Knew It →
What midpoint formula gives the center of a circle from diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
What midpoint formula gives the center of a circle from diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
Tap to reveal answer
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Midpoint formula gives center from endpoints.
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Midpoint formula gives center from endpoints.
← Didn't Know|Knew It →
What is the radius in terms of diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
What is the radius in terms of diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
Tap to reveal answer
$r=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Radius is half the distance between endpoints.
$r=\frac{1}{2}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Radius is half the distance between endpoints.
← Didn't Know|Knew It →
What is the equation of the circle in standard form: $x^2+y^2=9$?
What is the equation of the circle in standard form: $x^2+y^2=9$?
Tap to reveal answer
$(x-0)^2+(y-0)^2=9$. Already in standard form with center $(0,0)$.
$(x-0)^2+(y-0)^2=9$. Already in standard form with center $(0,0)$.
← Didn't Know|Knew It →
What is the general form after expanding $(x-2)^2+(y+3)^2=16$?
What is the general form after expanding $(x-2)^2+(y+3)^2=16$?
Tap to reveal answer
$x^2+y^2-4x+6y-3=0$. Expand and rearrange to general form.
$x^2+y^2-4x+6y-3=0$. Expand and rearrange to general form.
← Didn't Know|Knew It →
What is the general form after expanding $(x+5)^2+(y-1)^2=36$?
What is the general form after expanding $(x+5)^2+(y-1)^2=36$?
Tap to reveal answer
$x^2+y^2+10x-2y-10=0$. Expand and rearrange to general form.
$x^2+y^2+10x-2y-10=0$. Expand and rearrange to general form.
← Didn't Know|Knew It →
What is the general form after expanding $x^2+(y-4)^2=20$?
What is the general form after expanding $x^2+(y-4)^2=20$?
Tap to reveal answer
$x^2+y^2-8y-4=0$. Expand and rearrange to general form.
$x^2+y^2-8y-4=0$. Expand and rearrange to general form.
← Didn't Know|Knew It →
Identify the value added to complete the square for $x^2-14x$.
Identify the value added to complete the square for $x^2-14x$.
Tap to reveal answer
$49$. Take half of coefficient: $(-14/2)^2=49$.
$49$. Take half of coefficient: $(-14/2)^2=49$.
← Didn't Know|Knew It →
Identify the value added to complete the square for $y^2+9y$.
Identify the value added to complete the square for $y^2+9y$.
Tap to reveal answer
$\frac{81}{4}$. Take half of coefficient: $(9/2)^2=81/4$.
$\frac{81}{4}$. Take half of coefficient: $(9/2)^2=81/4$.
← Didn't Know|Knew It →
What is the completed-square expression equivalent to $x^2-12x$?
What is the completed-square expression equivalent to $x^2-12x$?
Tap to reveal answer
$(x-6)^2-36$. Complete square: $(x-6)^2=x^2-12x+36$.
$(x-6)^2-36$. Complete square: $(x-6)^2=x^2-12x+36$.
← Didn't Know|Knew It →
What is the completed-square expression equivalent to $y^2+8y$?
What is the completed-square expression equivalent to $y^2+8y$?
Tap to reveal answer
$(y+4)^2-16$. Complete square: $(y+4)^2=y^2+8y+16$.
$(y+4)^2-16$. Complete square: $(y+4)^2=y^2+8y+16$.
← Didn't Know|Knew It →
What is the center and radius of $(x+1)^2+(y-2)^2=0$?
What is the center and radius of $(x+1)^2+(y-2)^2=0$?
Tap to reveal answer
Center $(-1,2)$; radius $0$. Degenerate circle: single point at $(-1,2)$.
Center $(-1,2)$; radius $0$. Degenerate circle: single point at $(-1,2)$.
← Didn't Know|Knew It →
What is the center and radius of $(x-7)^2+(y+3)^2=81$?
What is the center and radius of $(x-7)^2+(y+3)^2=81$?
Tap to reveal answer
Center $(7,-3)$; radius $9$. Read center and radius from standard form.
Center $(7,-3)$; radius $9$. Read center and radius from standard form.
← Didn't Know|Knew It →
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 $r$?
Tap to reveal answer
$(x-h)^2+(y-k)^2=r^2$. Standard form shows center $(h,k)$ and radius $r$ explicitly.
$(x-h)^2+(y-k)^2=r^2$. Standard form shows center $(h,k)$ and radius $r$ explicitly.
← Didn't Know|Knew It →
What is the center and radius of the circle $(x-h)^2+(y-k)^2=r^2$?
What is the center and radius of the circle $(x-h)^2+(y-k)^2=r^2$?
Tap to reveal answer
Center $(h,k)$; radius $r$. Read center and radius directly from standard form.
Center $(h,k)$; radius $r$. Read center and radius directly from standard form.
← Didn't Know|Knew It →
What is the equation of a circle centered at the origin with radius $r$?
What is the equation of a circle centered at the origin with radius $r$?
Tap to reveal answer
$x^2+y^2=r^2$. When center is at origin, $h=0$ and $k=0$.
$x^2+y^2=r^2$. When center is at origin, $h=0$ and $k=0$.
← Didn't Know|Knew It →