Standard form equation and general form equation?

Question

1) Write standard form equation and the general form of equation of circle with radius and center (h,k)
r=10 and (h,K)= (-6,-8)

2) Circle has equation x^2+y^2+2x-2y-34=0
graph circle (h,K) and radius and find intercepts

3) Find center (h,k) and radius of circle and find intercepts
2x^2+24x+2y^2=0

Please help show step by step directions

Answer 1

1. Standard form circle equation is (x - h)² + (y - k)² = r². wher (h, k ) is center and r is radius.

Substitute (h, k )   = (-6, -8) and r = 10 in the equation

(x - (-6))² + (y - (-8))² = 10²

Standard form circle equation is (x + 6)² + (y + 8)² = 10²

General form equation is x² + y² + Ax + By + C = 0

Expand the Standard form circle equation

x² + 12x + 36 + y² + 16y + 64 = 100

Group like terms

x² + y² + 12x +16y + 36 + 64 = 100

x² + y² + 12x +16y + 100 = 100

x² + y² + 12x +16y = 0

General form equation is x² + y² + 12x +16y = 0.

2. Standard form circle equation is (x - h)² + (y - k)² = r². wher (h, k ) is center and r is radius.

Circle equation is x²+y²+2x-2y-34=0

Write varables and constants a side.

x²+y²+2x-2y = 34

(x²+ 2x) + (y² -2y )= 34

Now complete the square.

First complete the square for x terms.

Add (half the x coefficent)² to each side.  

So, (half the x coefficent)² = (2/2)²= (1)² add to each side

(x²+ 2x + 1²) + (y² -2y ) = 34 + 1²

(x²+ 2x + 1²) + (y² -2y ) = 35

Now complete the square for y terms

Add (half the y coefficent)² to each side.  

So, (half the y coefficient)² = (2/2)²= (1)² add to each side

(x²+ 2x + 1²) + (y² -2y + 1² ) = 35 + 1²

(x²+ 2x + 1²) + (y² - 2y + 1² ) = 36

Simplify

(x +1)²+ (y - 1)²= 6²

(x - (-1)⁾²+ (y - 1))²= 6²

(h, k )= (-1, 1) and r = 6.

3. Standard form circle equation is (x - h)² + (y - k)² = r². wher (h, k ) is center and r is radius.

Circle equation is 2x²+24x+2y²=0

Make the x² and y² coeffciients 1 by dividing the equation by 2.

x²+ 12x + y²=0

(x²+ 12x) + (y² - 0 )= 0

 Now complete the square.

First complete the square for x terms.

Add (half the x coefficent)² to each side.  

So, (half the x coefficent)² = (12/2)²= (6)² add to each side

(x²+ 12x + 6²) + (y² - 0 ) =   6²

Simplify

(x   + 6)²+ (y - 0)²= 6²

(x   - (-6))²+ (y - 0)²= 6²

(h, k )= (-6, 0) and r = 6.

 

Comments

• To find x - intercept, substitute y = 0 in (x + 6)²+ y²= 36^.

(x + 6)² + 0² = 36

(x + 6)² = 36

x + 6 = ± 6

⇒ x = ± 6 - 6.

So, the x - intercepts are 0 and - 12.

To find y - intercept, substitute x = 0 in (x + 6)²+ y²= 36.

(0 + 6)² + y² = 36

36 + y² = 36

y²= 36 - 36 = 0

⇒ y = 0.

So, the y - intercept is 0.

Answer 2

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 + 2x - 2y - 34 = 0.

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)² to each side of the expression.

Here, x coefficient = 2, so, (half the x coefficient)² = (2/2)²= 1.

Here, y coefficient = - 2, so, (half the y coefficient)² = (- 2/2)²= 1.

Add 1 and 1 to each side.

x² + 2x + 1 + y² - 2y + 1 = 34 + 1 + 1

(x + 1)² + (y - 1)² = 36

(x - (- 1))² + (y - 1)² = 6²

Compare the equation with standard form of a circle equation.

• The center (h, k) is (- 1, 1) and
• The radius (r) is 6 units.
• To find x - intercept, substitute y = 0 in (x + 1)² + (y - 1)² = 36

(x + 1)² + (0 - 1)² = 36

(x + 1)² + 1 = 36

(x + 1)²= 36 - 1 = 35

x + 1 = ± √35

⇒ x = ± √35 - 1.

So the x - intercepts are 4.916 and - 6.916.

To find y - intercept, substitute x = 0 in (x + 1)² + (y - 1)² = 36

(0 + 1)² + (y - 1)² = 36

1 + (y - 1)² = 36

(y - 1)²= 36 - 1 = 35

y - 1 = ± √35

⇒ y = ± √35 + 1.

So the y - intercepts are 6.916 and - 4.916.

• GRAPH :

1. Draw the coordinate plane.

2. Place the center of the circle at (- 1, 1).

3. Plot the radius points on the coordinate plane.

   Since, the radius is 6 units,

  Count 6 units up, down, left, and right from the center (- 1, 1).

  This means that,

  Up (- 1, 1 + 6) = (- 1, 7)

  Down (- 1, 1 - 6) = (- 1, - 5)

  Left (- 1 - 6, 1) = (- 7, 1)

  Right (- 1 + 6, 1) = (5, 1)

The circle should has the points at (- 1, 7), (- 1, - 5), (- 7, 1), and (5, 1).

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