Graph the circle 4x^2+4y^2-24x-16y-29=0
You need the centre and the radius.
complete the square
4(x^2 - 6x + .....) + 4(y^2 - 4y + ....) = 29
4(x^2 - 6x + 9) + 4(y^2 - 4y + 4) = 29 + 36 + 16
4(x-3)^2 + 4(y-2)^2 = 81
divide by 4
(x-3)^2 + (y-2)^2 = 81/4
so the radius is 9/2 and the centre is (3,2)
To graph the circle given by the equation 4x^2+4y^2-24x-16y-29=0, we can complete the square to write the equation in standard form.
First, let's group the x and y terms:
4x^2 - 24x + 4y^2 - 16y = 29
Next, we'll complete the square for the x-terms. To do this, we need to add and subtract the square of half the coefficient of x:
4(x^2 - 6x) + 4y^2 - 16y = 29
Now, we add and subtract (6/2)^2 = 9 to complete the square inside the parentheses:
4(x^2 - 6x + 9) + 4y^2 - 16y = 29 + 4(9)
Simplifying this equation gives:
4(x - 3)^2 + 4y^2 - 16y = 65
Next, we'll complete the square for the y-terms. We'll add and subtract the square of half the coefficient of y:
4(x - 3)^2 + 4(y^2 - 4y) = 65
Similarly, we add and subtract (4/2)^2 = 4 to complete the square inside the parentheses:
4(x - 3)^2 + 4(y^2 - 4y + 4) = 65 + 4(4)
Simplifying this equation further results in:
4(x - 3)^2 + 4(y - 2)^2 = 81
Finally, we divide both sides of the equation by 4 to isolate the squares:
(x - 3)^2 + (y - 2)^2 = 81/4
Our equation is now in standard form, (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius.
From the equation, we can determine that the center of the circle is at (3, 2) and the radius is sqrt(81/4) = 9/2.
To graph this circle, we plot the center at (3, 2) and draw a circle with a radius of 9/2 around it.
To graph the circle represented by the equation 4x^2 + 4y^2 - 24x - 16y - 29 = 0, we can follow a few steps:
Step 1: Rearrange the equation to the standard form of a circle.
4x^2 + 4y^2 - 24x - 16y - 29 = 0
Divide each term by 4 to simplify the equation:
x^2 + y^2 - 6x - 4y - 29/4 = 0
Step 2: Complete the square for both variables to get the equation into a centered form.
- For x: Take half of the coefficient of x (-6) and square it, then add both sides of the equation:
x^2 - 6x + (-6/2)^2 = 6^2/4
x^2 - 6x + 9 = 36/4
- For y: Take half of the coefficient of y (-4) and square it, then add both sides of the equation:
y^2 - 4y + (-4/2)^2 = 4^2/4
y^2 - 4y + 4 = 16/4
The equation becomes:
(x - 3)^2 + (y - 2)^2 = 45/4
Step 3: Compare the equation to the standard equation of a circle to find the center and radius.
The standard equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Comparing it to our equation:
(x - 3)^2 + (y - 2)^2 = 45/4
The center of the circle is (h, k) = (3, 2).
The radius r^2 = 45/4, so the radius r ≈ √(45/4) ≈ 3.35 units.
Step 4: Plot the center and the radius on the coordinate plane.
The center of the circle is at (3, 2), so mark this point on the graph.
The radius of the circle is approximately 3.35 units, so draw a circle around the center with a radius of 3.35 units.
Step 5: Label the graph as a circle with the center and radius.
You can label the graph as "Circle with center (3, 2) and radius ≈ 3.35 units."
Now, you have successfully graphed the circle represented by the equation 4x^2 + 4y^2 - 24x - 16y - 29 = 0.