In standard form. Find the center, radius, intercepts, and graph the circle. x^2+y^2-4x+18y=-69

Question

Please help? I'm confused with how the x and y numbers are different. Can someone show and explain to me how this problem is done? Thanks.

Answer 1

x^2+y^2-4x+18y = -69

Add 69 to each side.

x^2+y^2-4x+18y+69 = -69+69

x^2+y^2-4x+18y+69 = 0 ------> (1)

Compare it to circle equation x^2+y^2+2gx+2fy+c = 0

2g = -4, 2f = 18, c = 69

g = -4/2, f = 18/2

g = -2, f = 9

Center = (-g,-f) = (-(-2), -9) = (2,-9)

r = √(g^2+f^2-c)

r = √((-2)^2+(9)2-69)

r = √(4+81-69)

r = √16

r = 4

Radius of circle is 4

.We can also write standard equation (x-h)^2+(y-k)^2 = r^2

(x -2)^2+(y+9)^2 =16. ---------> (1)

To find x intercept  substitute y = 0 in below equation (1).

(x -2)^2+(y+9)^2 =16.

(x-2)^2+81 = 16

(x-2)^2 = 16-81

(x-2)^2 = -73

(x-2) = √-73

x = 2+√-73

Squre root negitive is not possible, so no x inetercept here.

To find y intercept  substitute x = 0 in equation (1).

(x -2)^2+(y+9)^2 =16.

4+(y+9)^2 = 16

(y+9)^2 = 16-4

y+9 = +√ 12     and     y+9 = -√ 12

y = -9+√ 12     and      y = -9-√ 12

y = -9+3.46     and     y = -9-3.46

y = -5.53      and      y = -12.46

Intercepts are (0,-5.5) ,(0,-12.4).

Answer 2

The standard form of the circle equation is ( x - h )² + ( y - k )² = r², where, (h, k) is the center of the circle, and r is the radius.

The equation is x² + y² - 4x + 18y = - 69.

Write the equation in standard form of a circle.

To change the expression into a perfect square  add (half the x coefficient)² and add (half the y coefficient)²to each side of the expression.

Here x coefficient = - 4, so, (half the x coefficient)² = (- 4/2)²= 4.

Here y coefficient = 18, so, (half the y coefficient)² = (18/2)²= 81.

Add 4 and 81 to each side.

x² - 4x + 4 + y² + 18y + 81 = - 69 + 4 + 81

(x - 2)² + (y + 9)² = 16

(x - 2)² + (y - (- 9))² = 4²

Compare the equation with standard form of a circle equation.

• The center (h, k) is (2, - 9) and
• The radius (r) is 4 units.
• To find x -intercept, substitute y = 0 in (x - 2)² + (y + 9)² = 16

(x - 2)² + (0 + 9)² = 16

(x - 2)² + 9² = 16

(x - 2)²= 16 - 81 = - 65

x - 2 = ± √(- 65)

⇒ x = ± √(- 65) + 2.

Negative square root is imaginary,So there is no x- intercept.

To find y - intercept, substitute x = 0 in (x - 2)² + (y + 9)² = 16

(0 - 2)² + (y + 9)² = 16

(- 2)² + (y + 9)² = 16

(y + 9)²= 16 - 4 = 12

y + 9 = ± √12

⇒ y = ± 2√3 - 9.

So the y - intercepts are - 5.53 and - 12.46.

• GRAPH :

1. Draw the coordinate plane.

2. Place the center of the circle at (2, - 9).

3. Plot the radius points on the coordinate plane.

   Since radius is 4 units,

  Count 4 units up, down, left, and right from the center (2, - 9).

  This means that we should have the points at (2, - 5), (2, - 13), (- 2, 9), and (6, 9).

4. Connect the plotted points to the graph of the circle with a round, smooth  curve.