Wize High School Grade 12 Pre-Calculus Textbook > Conics [BC/AB]

Circles

Circles 
A circle is:
a collection of points that are the same distance from a single point
generated when a plane is perpendicular to the axis of the cone
not a function
can be expressed as (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2(x−h)2+(y−k)2=r2
(h,k)(h,k)(h,k) is the center
rrr is the radius

Wize Tip
Helpful formulas for this section are:

∣xy∣=(x2−x1)2+(y2−y1)2|xy|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}∣xy∣=(x2​−x1​)2+(y2​−y1​)2​

Midppoint=(x2+x12,y2+y12)M_{idppoint}=\Bigg(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\Bigg)Midppoint​=(2x2​+x1​​,2y2​+y1​​)

Example: Circles 
Identify the center and the radius of the circle 4x2+4y2−8x+16y−300=04x^2+4y^2-8x+16y-300=04x2+4y2−8x+16y−300=0.

4x2+4y2−8x+16y−300=0(4x2−8x)+(4y2+16y)=3004(x2−2x)+4(y2+4y)=3004(x2−2x+1)+4(y2+4y+4)=300+4+164(x−1)2+4(y+2)2=320(x−1)2+(y+2)2=80\begin{array}{rcl}
4x^2+4y^2-8x+16y-300&=&0\\\\
(4x^2-8x)+(4y^2+16y)&=&300\\\\
4(x^2-2x)+4(y^2+4y)&=&300\\\\
4(x^2-2x+1)+4(y^2+4y+4)&=&300+4+16\\\\
4(x-1)^2+4(y+2)^2&=&320\\\\
(x-1)^2+(y+2)^2&=&80
\end{array}4x2+4y2−8x+16y−300(4x2−8x)+(4y2+16y)4(x2−2x)+4(y2+4y)4(x2−2x+1)+4(y2+4y+4)4(x−1)2+4(y+2)2(x−1)2+(y+2)2​======​0300300300+4+1632080​


The center is at (1,−2)(1,-2)(1,−2) and the radius is 80=45\sqrt{80}=4\sqrt{5}80​=45​.

Practice: Circles 
Write the equation of a circle with radius 21\sqrt{21}21​ and center (0,9)(0,9)(0,9).

A) x2+(y−9)2=21x^2+(y-9)^2=\sqrt{21}x2+(y−9)2=21​ 

B) x2+(y−9)2=21x^2+(y-9)^2=21x2+(y−9)2=21 

C) (x−9)2+y2=21(x-9)^2+y^2=21(x−9)2+y2=21 

D) (x−9)2+y2=21(x-9)^2+y^2=\sqrt{21}(x−9)2+y2=21​ 

I don't know

Practice: Circles 
What is the center and the radius of the circle 9x2+9y2+18x+18y−81=09x^2+9y^2+18x+18y-81=09x2+9y2+18x+18y−81=0?

a. Center

b. Radius

I don't know

Practice: Circles 
A circle has a diameter containing the point (−12,3)(-12, 3)(−12,3) and (10,1)(10,1)(10,1). 

a. What is the radius? Write in exact, simplified form.

b. Where is the center?

I don't know

Extra Practice

Circles

Practice: Circles 
A circle has a diameter containing the point (14,13)\Bigg(\dfrac{1}{4},\dfrac{1}{3}\Bigg)(41​,31​) and (32,57)\Bigg(\dfrac{3}{2},\dfrac{5}{7}\Bigg)(23​,75​). 

Circles

Practice: Circles 
A circle has a diameter containing the point (5,−7)(5, -7)(5,−7) and (−8,13)(-8,13)(−8,13). 

Circles

Practice: Circles 
What is the center and the radius of the circle x2+5x+y2−y=52x^2+5x+y^2-y=\dfrac{5}{2}x2+5x+y2−y=25​?

Circles

Practice: Circles 
What is the center and the radius of the circle x2+6x+y2+4y=8x^2+6x+y^2+4y=8x2+6x+y2+4y=8?

Circles

Practice: Circles 
Write the equation of a circle with radius 101010 and center (12,54)\Bigg(\dfrac{1}{2},\dfrac{5}{4}\Bigg)(21​,45​).

Circles

Practice: Circles 
Write the equation of a circle with radius 54\sqrt{54}54​ and center (−3,5)(-3,5)(−3,5).

Wize High School Grade 12 Pre-Calculus Textbook > Conics [BC/AB]

Circles

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Circles

Circles

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Circles

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