write the equation of the circle in standard form. find the center, radius, intercepts, and graph the circle. x^2+y^2-6x-8y+25=36
x ^ 2 + y ^ 2 - 6 x - 8 y + 25 = 36
Equation of a circle in standard form :
( x - h ) ^ 2 + ( y - k ) ^ 2 = r ^ 2
x ^ 2 + y ^ 2 - 6 x - 8 y + 25 = 36
x ^ 2 - 6 x + y ^ 2 - 8 y + 25 = 36
( x ^ 2 - 6 x ) + ( y ^ 2 - 8 y ) + 25 = 36
The process involves completing the square separately for the x and y variables.
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( x - 3 ) ^ 2 = x ^ 2 - 2 * x * 3 + 3 ^ 2
( x - 3 ) ^ 2 = x ^ 2 - 6 x + 9 Subtract 9 to both sides
( x - 3 ) ^ 2 - 9 = x ^ 2 - 6 x + 9 - 9
( x - 3 ) ^ 2 - 9 = x ^ 2 - 6 x
x ^ 2 - 6 x = ( x - 3 ) ^ 2 - 9
( y - 4 ) ^ 2 = y ^ 2 - 2 * y * 4 + 4 ^ 2
( y - 4 ) ^ 2 = y ^ 2 - 8 y + 16 Subtract 16 to both sides
( y - 4 ) ^ 2 - 16 = y ^ 2 - 8 y + 16 -16
( y - 4 ) ^ 2 - 16 = y ^ 2 - 8 y
y ^ 2 - 8 y = ( y - 4 ) ^ 2 - 16
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( x ^ 2 - 6 x ) + ( y ^ 2 - 8 y ) + 25 = 36
( x - 3 ) ^ 2 - 9 + ( y - 4 ) ^ 2 - 16 + 25 = 36
(x - 3 ) ^ 2 + ( y - 4 ) ^ 2 - 25 + 25 = 36
( x - 3 ) ^ 2 + ( y - 4 ) ^ 2 = 36
( x - 3 ) ^ 2 + ( y - 4 ) ^ 2 = 6 ^ 2
To write the equation of the circle in standard form and find its center, radius, and intercepts, we can rearrange the given equation to the standard form equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2.
The equation given is: x^2 + y^2 - 6x - 8y + 25 = 36.
Step 1: Bringing the constant term to the right side:
x^2 + y^2 - 6x - 8y = 36 - 25
Simplifying the equation gives:
x^2 + y^2 - 6x - 8y = 11
Step 2: Completing the square for both x and y terms:
For the x terms:
To complete the square for x, we take half of the coefficient of x, square it, and add it to both sides of the equation. The coefficient of x is -6, so (-6/2)^2 = (-3)^2 = 9.
Adding 9 to both sides gives:
x^2 - 6x + 9 + y^2 - 8y = 11 + 9
Now, let's complete the square for the y terms:
The coefficient of y is -8, so (-8/2)^2 = (-4)^2 = 16.
Adding 16 to both sides gives:
x^2 - 6x + 9 + y^2 - 8y + 16 = 11 + 9 + 16
Simplifying further gives:
(x - 3)^2 + (y - 4)^2 = 36
The equation is now in the standard form of a circle. We can identify the center, radius, and intercepts from this.
Step 3: Determining the center and radius:
Comparing the equation to the standard form equation (x - h)^2 + (y - k)^2 = r^2, we can see that the center of the circle is at (h, k), which in this case is (3, 4). The radius of the circle is the square root of the constant term on the right side, which is 6 in this case.
Center: (3, 4)
Radius: 6
Step 4: Finding the intercepts:
For the x-intercepts, we set y = 0 and solve for x:
(x - 3)^2 + (0 - 4)^2 = 36
(x - 3)^2 + 16 = 36
(x - 3)^2 = 20
Taking the square root of both sides:
x - 3 = ±√20
x - 3 = ±2√5
Solving for x gives two x-intercepts:
x = 3 + 2√5
x = 3 - 2√5
For the y-intercepts, we set x = 0 and solve for y:
(0 - 3)^2 + (y - 4)^2 = 36
9 + (y - 4)^2 = 36
(y - 4)^2 = 27
Taking the square root of both sides:
y - 4 = ±√27
y - 4 = ±3√3
Solving for y gives two y-intercepts:
y = 4 + 3√3
y = 4 - 3√3
Step 5: Graphing the circle:
To graph the circle, plot the center at (3, 4) and draw a circle with a radius of 6.
Here's a visual representation of the circle:
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