Write equation in standard form. Find center, radius, intercepts, and graph the circle.

Question

Write the equation of the circle in standard form. Find the center, radius, intercepts, and graph the circle. x^2+y^2+10x−18y=−102

Answer 1

Standard form of a circle (x - h)^2+(y - k)^2 = r^2

Here r represents radious of circle and center point is(h, k)

Given equation x^2+y^2+10x-18y = -102 ---------- >(1).

x^2+y^2+2*5x-2*9y = -102

Add 102 to each side.

x^2+2*5x+y^2-2*9y+102 = -102+102

x^2+2*5x+y^2-2*9y+102 = 0

Add 4 to each side.

x^2+2*5x+y^2-2*9y+102+4 = 0+4

x^2+2*5x+y^2-2*9y+81+25 = 4

x^2+2*5x+25+y^2-2*9y+81 = 4

(x+5)^2+(y-9)^2 = 2^2

(x-(-5))^2+(y-9)^2 = 2^2

Compare standard form of circle.

h = -5, k = 9 and r = 2

center = (-5, 9) and radious = 2

Substitute x = 0 in equation (1)

0^2+y^2+10*0-18y = -102

y^2-18y+102 = 0

roots of above quadratic equation = [-b ± √(b^2-4ac)]/2a

y = [-(-18)± √(324-408)]/2

y = [18± √(-84)]/2

y = [18± √(4*-21)]/ 2

y = [18 ±2√(-21)]/2

Take common out 2

y =2 [9± √(-21)]/2

y = 9± √(-21)

Negitive square root is imaginary,so no y intercept here.

Substitute y = 0 in equation (1).

x^2+0^2+10x-18*0 = -102

x^2+10x+102 = 0

x = [-10± √(100-408)]/2

x = [-10± √(-308)]/2

x = [-10± √(4*-77)]/2

x = [-10± 2√(-77)]/2

Take common out 2

x = 2[-5± √(-77)]/2

x = -5± √(-77)

Negitive square root is imaginary, so no x intercept here.

Answer 2

Standard form of a circle equation is

Where center is (h ,k ) and radius of circle is r .

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 = 10. so, (half the x coefficient)² = (10/2)²= 25

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

Add 25 and 81 to each side

----> (1)

Compare it to standard form of circle

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

Negative squre root is imaginary,So there is no y  - intercept.

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

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

Center at (-5,9) and r = 2,So we plot

Up (-5, 9+2) = (-5, 11)

Down (-5, 9-2) = (-5, 7)

Left (-5-2, 9) = (-7, 9)

Right (-5+2, 9) = (-3, 9)

Graph

1. draw the coordinate plane.

2. Plot the center at (-5,9).

3. Plot 4 points " radius away from the center in the up, down, left and right direction.

4. Sketch the circle.