Question 202800: 3x^2+3y^2-24x-12y-15=0 Graph the circle using (h,k) and R find the intercepts if any Found 2 solutions by jsmallt9, Earlsdon:Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! To find (h, k) and R we need to manipulate the equation so it is in the standard form for circles: . This requires that we have the sum of two perfect squares on the left. So we need to "complete the squares" on the left side so we can rewrite them as (x - h)^2 and (y - k)^2.
To complete the squares:
Gather all the variable terms on one side (and all the constant terms on the other side:
Subtract 15 from both sides
Change subtractions, if any, into additions so you can rearrange the terms (using the Commutative Property) and get the x terms and the y terms together:
Since the coefficients of the x^2 and Y^2 terms are the same, divide both sides by this coefficient.
Now comes the tricky part. Complete the square for each variable by
Figuring out what 1/2 of the coefficient of the first power term is.
For the x's: 1/2 of -8 is -4
For the y's: 1/2 of -4 is -2
Squaring this "1/2 of the coefficient.
For the x's' (-4)^2 = 16
For the y's' (-2)^2 = 4
Add these squares to both sides of the equation. On the left side, add the squares after the terms you used to calculate them.
Since we want (x - h)^2 and (y - k)^2, we'll rewrite the equation with subtractions.
Both the first three terms on the left and the last three terms on the left fit the perfect square pattern:
So we now rewrite each set of three terms as the perfect squares they are:
The only thing left to do is to write the right side as a perfect square:
Now that we have transformed
into
which is in the standard form (for circles) of (x - h)^2 + (y - k)^2 = r^2 we can read the center: (4, 2) and the radius: 5 and use these to graph the circle.
You can put this solution on YOUR website! Divide both sides by 3. Group the x- and y-terms as shown: Add 5 to both sides. Complete the square in both the x-terms and the y-terms. Factor the left groups ans simplify the right side. Compare with the standard form for a circle with center at (h, k) and radius r.
The center is at (4, 2) and the radius is 5.
To graph this circle, we will have to solve this equation for y and the graph each of the two solutions separately. Subtract from both sides. Simplify the right side. Take the square root of both sides. The right side should have a + or - sign in front. Add 2 to both sides. Now graph both of these solutions.
The y-intercepts are: (0, 5) and (0, -1)
The x-intercepts are: (8.58, 0) and (-0.58, 0)
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