Circles equations

Find the missing value to complete the square. 6.x 2 – 2x +7. x 2 + 4x +8. x 2 – 6x + Circles – Warm Up Find the missing value to complete the square. 6.x 2 – 2x +7. x 2 + 4x +8. x 2 – 6x + Simplify

Solutions 6.x 2 – 2x + ; c = = – = (–1) 2 = 1 7.x 2 + 4x + ; c = = = 2 2 = 4 8.x 2 – 6x + ; c = = – = (–3) 2 = 9 b2b2 b2b2 b2b b2b2 b2b2 b2b2 b2b = = = 4  5 = = 16  3 = = 36  2 = 6 2

CIRCLE TERMS EQUATION FORM CENTER RADIUS MIDPOINT FORMULA DISTANCE FORMULA (x – h)² + (y – k)² = r² (h, k ) r r C=(h, k) Definition: A circle is an infinite number of points a set distance away from a center

Write an equation of a circle with center (3, –2) and radius 3. Circles (x – h) 2 + (y – k) 2 = r 2 Use the standard form of the equation of a circle. (x – 3) 2 + (y – (–2)) 2 = 3 2 Substitute 3 for h, –2 for k, and 3 for r. (x – 3) 2 + (y + 2) 2 = 9Simplify. Check:Solve the equation for y and enter both functions into your graphing calculator. (x – 3) 2 + (y + 2) 2 = 9 (y + 2) 2 = 9 – (x – 3) 2 y + 2 = ± 9 – (x – 3) 2 y = –2 ± 9 – (x – 3) 2

Write an equation for the translation of x 2 + y 2 = 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 2 Use 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 the equation of a circle in standard form A) write the equation of the circle in standard form-steps Group the x and y terms Move the constant term Complete the square for x, y Put in standard form Ex: x² + y² - 4x + 8y + 11 = 0

WRITE and GRAPH A) write the equation of the circle in standard form x² + y² - 4x + 8y + 11 = 0 Group the x and y terms x² - 4x + y² + 8y + 11 = 0 Complete the square for x, y x² - 4x y² + 8y + 16 = (x – 2)² + (y + 4)² = 9 B) GRAPH Plot Center (2,-4) Radius = 3

WRITE and then GRAPH A) write the equation of the circle in standard form 4x² + 4y² + 36y + 5 = 0 Group terms, move constant, factor out the coefficient. Complete the square for x,y Divide both sides by coefficient

WRITE and then GRAPH A) write the equation of the circle in standard form 4x² + 4y² + 36y + 5 = 0 Group terms, move constant, factor out the coefficient. 4(x²) + 4(y² + 9y + __ ) = -5 + ___ Notice that x is already done!

WRITE and GRAPH A) write the equation of the circle in standard form 4x² + 4y² + 36y + 5 = 0 Complete the square for y: 4(x²) + 4(y² + 9y+__) = -5 +__ 4(x²)+4(y² + 9y + 81/4) = (x²) + 4(y + 9/2)² = 76 Divide by 4: x² + (y + 9/2)² = 19 Center (0, -9/2) Radius =

WRITING EQUATIONS Write the EQ of a circle that has a center of (-5,7) and passes through (7,3) Plot your info Need to find values for h, k, r (h, k) = (, ) How do we find r? Use distance form. for C and P. Plug into circle formula (x – h)² + (y – k)² = r² C = (-5,7) P = (7,3) radius

WRITING EQUATIONS Write the EQ of a circle that has a center of (-5,7) and passes through (7,3) Plot your info Need to find values for h, k, r (h, k) = (-5, 7) How do we find r? Use distance form. for C and P. Plug into circle formula (x – h)² + (y – k)² = r² (x + 5)² + (y – 7)² = (4√10)² (x + 5)² + (y – 7)² = 160 C = (-5,7) P = (7,3) radius

Let’s Try One Write 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) radius Plot your info Need to find values for h, k, r How do we find (h,k)? Use midpoint formula (h, k) = How do we find r? Use dist form with C and B. Plug into formula (x – h)² + (y – k)² = r² Hint: Where is the center? How do you find it?

Let’s Try One Write 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) radius Plot your info Need to find values for h, k, r How 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)² + (y – k)² = r² (x)² + (y + 2)² = 32 Hint: Where is the center? How do you find it?

Suppose the equation of a circle is (x – 5)² + (y + 2)² = 9 Write the equation of the new circle given that: A) The center of the circle moved up 4 spots and left 5: (x – 0) ² + (y – 2)² = 9 Center moved from (5,-2)  (0,2) B) The center of the circle moved down 3 spots and right 6: (x – 11) ² + (y + 5)² = 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. Let‘s Try One The center of the circle is (–4, 2). The radius is 6. (x – h) 2 + (y – k) 2 = r 2 Use the standard form. (x + 4) 2 + (y – 2) 2 = 36Write the equation. (x – (–4)) 2 + (y – 2) 2 = 6 2 Rewrite the equation in standard form. h = –4 k = 2 r = 6Find h, k, and r.

Graph (x – 3) 2 + (y + 1) 2 = 4. Let’s Try One (x – h) 2 + (y – k) 2 = r 2 Find the center and radius of the circle. (x – 3) 2 + (y – (–1)) 2 = 4 h = 3 k = –1 r 2 = 4, or r = 2 Draw the center (3, –1) and radius 2. Draw a smooth curve.

Solving linear quadratic systems Do now: page 182 # 37

Solving a circle-line system Y=-x-2 Plot the line b=-2 m = -1 Plot the center and count in four directions for the radius Find the points of intersection y=-x -2 x² + (y + 2)² = 32

Solving a circle-line system (-4,2) (4,-6) Y=-x-2 Plot the line b=-2 m = -1 Plot the center and count in four directions for the radius Find the points of intersection y=-x -2 x² + (y + 2)² = 32

Solving a circle-line system By algebraic method: substitute –x-2 for y into second equation y=-x -2 x² + (y + 2)² = 32

Solving a circle-line system By algebraic method: substitute –x-2 for y into second equation y=-x -2 x² + (y + 2)² = 32 x² + (-x )² = 32 x² + ( -x)² = 32 2x 2 =32 X 2 =16 X=4, x=-4 solve for y: Y=-4-2y=4-2 Y=-6 (4,-6)y=2 (-4, 2)

Lets solve this one both ways