10.3 - Circles

Find the missing value to complete the square.

6.x2 2x +7. x2 + 4x +8. x2 6x +Circles Warm UpFind the missing value to complete the square.

6.x2 2x +7. x2 + 4x +8. x2 6x +Simplify.

1. 162. 493. 20 4. 485. 72

CIRCLE TERMS(x h) + (y k) = r (h, k )rrC=(h , k)Definition: A circle is an infinite number of points a set distance away from a center

EQUATION FORMCENTER RADIUSMIDPOINTFORMULADISTANCEFORMULA

Write an equation of a circle with center (3, 2) and radius 3.Circles (x h)2 + (y k)2 = r2Use the standard form of the equation of a circle.(x 3)2 + (y (2))2 = 32Substitute 3 for h, 2 for k, and 3 for r. (x 3)2 + (y + 2)2 = 9Simplify.

Write an equation for the translation of x2 + y2 = 16 two units right and one unit down.Circles(x 2)2 + (y (1))2 = 16Substitute 2 for h, 1 for k, and 16 for r 2. (x h)2 + (y k)2 = r 2Use the standard form of the equation of a circle. (x 2)2 + (y + 1)2 = 16Simplify. The equation is (x 2)2 + (y + 1)2 = 16.

WRITE and GRAPHA) write the equation of the circle in standard formx + y - 4x + 8y + 11 = 0Group the x and y terms x - 4x + y + 8y + 11 = 0Complete the square for x/y x - 4x + 4 + y + 8y + 16 = -11 + 4 + 16(x 2) + (y + 4) = 9YAY! Standard Form!

B) GRAPHPlot Center (2,-4)Radius = 3

WRITE and GRAPHA) write the equation of the circle in standard form4x + 4y + 36y + 5 = 0Group the x and y terms 4x + 4y + 36y + 5 = 0Complete the square for x/y4x + 4(y + 9y) = -54x + 4(y + 9y + 81/4) = -5 + 81 4x + 4(y + 9/2) = 76x + (y + 9/2) = 19YAY! Standard Form!

B) GRAPHPlot Center (0 , -9/2)Radius = 19 = 4.5

WRITING EQUATIONSWrite the EQ of a circle that has a center of (-5,7) and passes through (7,3)Plot your infoNeed to find values for h, k, and r(h , k) = (-5 , 7)How do we find r?Use distance formula with C and P.

Plug into formula(x h) + (y k) = r(x + 5) + (y 7) = (410)(x + 5) + (y 7) = 160

C = (-5,7) P = (7,3) radius

Lets Try OneWrite the EQ of a circle that has endpoints of the diameter at (-4,2) and passes through (4,-6)A = (-4,2) B = (4,-6) radiusPlot your infoNeed to find values for h, k, and rHow do we find (h,k)?Use midpoint formula

(h , k) = (0 , -2)How do we find r?Use dist form with C and B.

Plug into formula(x h) + (y k) = r(x) + (y + 2) = 32

Hint: Where is the center? How do you find it?

Suppose the equation of a circle is (x 5) + (y + 2) = 9Write the equation of the new circle given that:A) The center of the circle moved up 4 spots and left 5:(x 0) + (y 2) = 9 Center moved from (5,-2) (0,2)B) The center of the circle moved down 3 spots and right 6:(x 11) + (y + 5) = 9 Center moved from (5,-2) (11,-5)

Find the center and radius of the circle with equation (x + 4)2 + (y 2)2 = 36.Lets Try OneThe center of the circle is (4, 2). The radius is 6. (x h)2 + (y k)2 = r 2Use the standard form. (x + 4)2 + (y 2)2 = 36Write the equation. (x (4))2 + (y 2)2 = 62Rewrite the equation in standard form. h = 4 k = 2 r = 6Find h, k, and r.

Graph (x 3)2 + (y + 1)2 = 4.Lets Try One (x h)2 + (y k)2 = r 2Find the center and radius of the circle. (x 3)2 + (y (1))2 = 4h = 3 k = 1 r 2 = 4, or r = 2

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