Conics - Circle - Conic Sections

Conics - Circle

Basic concepts:
Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Midpoint formula: $M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)$
Conics - Ellipse

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Notes:

The conics form of a circle with center $\left( {h,\;k} \right)$ and radius $r$ is:
${\left( {\frac{{x - h}}{r}} \right)^2} + {\left( {\frac{{y - k}}{r}} \right)^2} = 1$

1.
graphing a circle
Sketch each circle and state the:
i) center
ii) radius
2.
converting a circle equation to conics form by “completing the square”
$4{x^2} + 4{y^2} + 24x - 8y + 15 = 0$
3.
finding the equation of a circle given its properties
Find the equation of a circle with:

Conics - Circle

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AlgebraDistance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=√​(x​2​​−x​1​​)​2​​+(y​2​​−y​1​​)​2​​​​​

AlgebraMidpoint formula: M=(x1+x22,y1+y22)M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)M=(​2​​x​1​​+x​2​​​​,​2​​y​1​​+y​2​​​​)

AlgebraConics - Ellipse

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Distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}d=√​(x​2​​−x​1​​)​2​​+(y​2​​−y​1​​)​2​​​​​

Midpoint formula: M=(x1+x22,y1+y22)M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)M=(​2​​x​1​​+x​2​​​​,​2​​y​1​​+y​2​​​​)

Conics - Ellipse

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Notes:

The conics form of a circle with center (h,k)\left( {h,\;k} \right)(h,k) and radius rrr is: (x−hr)2+(y−kr)2=1{\left( {\frac{{x - h}}{r}} \right)^2} + {\left( {\frac{{y - k}}{r}} \right)^2} = 1(​r​​x−h​​)​2​​+(​r​​y−k​​)​2​​=1

1.

graphing a circle Sketch each circle and state the: i) center ii) radius

a)

(x−53)2+(y+43)2=1{\left( {\frac{{x - 5}}{3}} \right)^2} + {\left( {\frac{{y + 4}}{3}} \right)^2} = 1(​3​​x−5​​)​2​​+(​3​​y+4​​)​2​​=1

b)

x2+y2=25{x^2} + {y^2} = 25x​2​​+y​2​​=25

2.

converting a circle equation to conics form by “completing the square” 4x2+4y2+24x−8y+15=04{x^2} + 4{y^2} + 24x - 8y + 15 = 04x​2​​+4y​2​​+24x−8y+15=0

a)

Convert the equation to conics form.

b)

Sketch the graph and state the: i) center ii) radius

3.

finding the equation of a circle given its properties Find the equation of a circle with:

a)

center (−3,5)\left( { - 3,\;5} \right)(−3,5), radius = 7

b)

center (2,0)\left( {2,\;0} \right)(2,0), passing through the point (−1,4)\left( { - 1,\;4} \right)(−1,4)

c)

diameter with endpoints (−9,4)\left( { - 9,\;4} \right)(−9,4) and (15,−6)\left( {15,\; - 6} \right)(15,−6)

d)

center (−2,−1)\left( { - 2,\; - 1} \right)(−2,−1), tangent to the line 3x+4y=153x + 4y = 153x+4y=15

Conics - Circle

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Algebra
Midpoint formula: $M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)$

Algebra
Conics - Ellipse

Distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Midpoint formula: $M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)$

or

The conics form of a circle with center $\left( {h,\;k} \right)$ and radius $r$ is:
${\left( {\frac{{x - h}}{r}} \right)^2} + {\left( {\frac{{y - k}}{r}} \right)^2} = 1$

graphing a circle
Sketch each circle and state the:
i) center
ii) radius

${\left( {\frac{{x - 5}}{3}} \right)^2} + {\left( {\frac{{y + 4}}{3}} \right)^2} = 1$

${x^2} + {y^2} = 25$

converting a circle equation to conics form by “completing the square”
$4{x^2} + 4{y^2} + 24x - 8y + 15 = 0$

Sketch the graph and state the:
i) center
ii) radius

finding the equation of a circle given its properties
Find the equation of a circle with:

center $\left( { - 3,\;5} \right)$ , radius = 7

center $\left( {2,\;0} \right)$ , passing through the point $\left( { - 1,\;4} \right)$

diameter with endpoints $\left( { - 9,\;4} \right)$ and $\left( {15,\; - 6} \right)$

center $\left( { - 2,\; - 1} \right)$ , tangent to the line $3x + 4y = 15$