Given the equation x^2+y^2-8x-6y=0
(a.) Write the equation in standard form .
(b.) State the center, radius, and intercepts.
(x^2 - 8x) + (y^2 - 6y) = 0
(x^2 - 8x + 16) + (y^2 - 6y + 9) = 16+9
(x-4)^2 + (y-3)^2 = 5^2
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To write the equation in standard form, we need to complete the square for both x and y terms.
(a.) Write the equation in standard form:
Starting with the given equation: x^2 + y^2 - 8x - 6y = 0
Rearranging the terms:
x^2 - 8x + y^2 - 6y = 0
Completing the square for x terms:
(x^2 - 8x) + y^2 - 6y = 0
Taking half of the coefficient of x (-8) and squaring it (16), we add it to both sides:
(x^2 - 8x + 16) + y^2 - 6y = 16
By applying the same procedure for y terms, we get:
(x^2 - 8x + 16) + (y^2 - 6y + 9) = 16 + 9
Factoring the completed squares:
(x - 4)^2 + (y - 3)^2 = 25
So, the equation in standard form is (x - 4)^2 + (y - 3)^2 = 25.
(b.) The standard form (x - h)^2 + (y - k)^2 = r^2 represents a circle with the center (h, k) and a radius of r.
From the standard form equation (x - 4)^2 + (y - 3)^2 = 25, we can determine the center, radius, and intercepts.
Center: The center of the circle is given by the opposite signs of the values inside the parentheses. So, (h, k) = (4, 3).
Radius: The radius is given by the square root of the constant term, which is 25 in this case. So, the radius, r, is √(25) = 5.
Intercepts: To find the x-intercepts, we set y = 0 and solve for x.
(x - 4)^2 + (0 - 3)^2 = 25
(x - 4)^2 + 9 = 25
(x - 4)^2 = 16
Taking the square root on both sides, we get:
x - 4 = ±4
x = 4 ± 4
So, x-intercepts are 8 and 0.
To find the y-intercepts, we set x = 0 and solve for y.
(0 - 4)^2 + (y - 3)^2 = 25
16 + (y - 3)^2 = 25
(y - 3)^2 = 9
Taking the square root on both sides, we get:
y - 3 = ±3
y = 3 ± 3
So, y-intercepts are 6 and 0.
Therefore, the center of the circle is (4, 3), the radius is 5, and the intercepts are (8, 0), (0, 6), and (0, 0).
To write the equation x^2 + y^2 - 8x - 6y = 0 in standard form, we need to complete the square for both x and y.
(a.) Writing the equation in standard form:
First, let's complete the square for x:
x^2 - 8x + ___ + y^2 - 6y = 0
To complete the square for x, we need to take half of the coefficient of x (-8), square it, and add it to both sides of the equation. Half of -8 is -4, and (-4)^2 is 16:
x^2 - 8x + 16 + y^2 - 6y = 16
Now, let's complete the square for y:
x^2 - 8x + 16 + y^2 - 6y + ___ = 16
To complete the square for y, we take half of the coefficient of y (-6), square it, and add it to both sides of the equation. Half of -6 is -3, and (-3)^2 is 9:
x^2 - 8x + 16 + y^2 - 6y + 9 = 16
Simplifying further:
(x^2 - 8x + 16) + (y^2 - 6y + 9) = 16
(x - 4)^2 + (y - 3)^2 = 16
Therefore, the equation in standard form is (x - 4)^2 + (y - 3)^2 = 16.
(b.) Now that we have the equation in standard form, we can determine the center, radius, and intercepts.
Center:
The center of the circle is given by the opposite sign of the coefficients of x and y in the equation. Thus, the center of the circle is (4, 3).
Radius:
The radius of the circle is obtained by taking the square root of the constant term on the right-hand side of the equation (16 in this case). The radius is √16 = 4.
Intercepts:
To find the x-intercepts, we set y = 0 and solve the equation:
(x - 4)^2 + (0 - 3)^2 = 16
(x - 4)^2 + 9 = 16
(x - 4)^2 = 7
Taking the square root of both sides, we get:
x - 4 = ±√7
x = 4 ± √7
So the x-intercepts are approximately x ≈ 4 + √7 and x ≈ 4 - √7.
Similarly, to find the y-intercepts, we set x = 0 and solve the equation:
(0 - 4)^2 + (y - 3)^2 = 16
16 + (y - 3)^2 = 16
(y - 3)^2 = 0
Taking the square root of both sides, we get:
y - 3 = 0
y = 3
Therefore, the y-intercept is y = 3.
In summary:
Center: (4, 3)
Radius: 4
X-intercepts: approximately x ≈ 4 + √7 and x ≈ 4 - √7
Y-intercept: y = 3