9x^2+25y^2-36x+250y+436=0 what is the center and the vertices of this ellipse?
To find the center and vertices of the ellipse represented by the equation 9x^2 + 25y^2 - 36x + 250y + 436 = 0, we need to rewrite the equation in a specific form called the standard form for an ellipse. The standard form of an ellipse equation is:
((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1
In this equation, (h, k) represents the coordinates of the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
To convert the given equation to standard form, follow these steps:
Step 1: Complete the squares for the x-terms and y-terms separately.
9x^2 - 36x + 25y^2 + 250y + 436 = 0
Step 2: Group the x-terms together and y-terms together.
(9x^2 - 36x) + (25y^2 + 250y) = -436
Step 3: Extract a common factor from each parentheses group.
9(x^2 - 4x) + 25(y^2 + 10y) = -436
Step 4: To complete the square for the x-terms, take half of the coefficient of x (which is -4), square it (which gives 16), and add it inside the parentheses. Do the same for the y-terms.
9(x^2 - 4x + 4) + 25(y^2 + 10y + 25) = -436 + 9(4) + 25(25)
Step 5: Simplify both sides of the equation.
9(x - 2)^2 + 25(y + 5)^2 = -436 + 36 + 625
9(x - 2)^2 + 25(y + 5)^2 = 225
Step 6: Divide both sides of the equation by 225 to isolate the equation to 1 on the right side.
[(x - 2)^2]/(225/9) + [(y + 5)^2]/(225/25) = 1
Step 7: Simplify the expressions in the denominators.
[(x - 2)^2]/25 + [(y + 5)^2]/9 = 1
Comparing this equation to the standard form of an ellipse, we can see that:
- The center of the ellipse is the point (h, k) = (2, -5).
- The length of the semi-major axis (a) is sqrt(25) = 5.
- The length of the semi-minor axis (b) is sqrt(9) = 3.
To find the vertices of the ellipse, we can use the equation:
(a^2 - b^2) = c^2, where c is the distance from the center to each vertex.
In this case, a = 5 and b = 3. Plugging these values into the equation, we get:
(5^2 - 3^2) = c^2
(25 - 9) = c^2
16 = c^2
Taking the square root of both sides, we find that c = 4.
Now, the vertices of the ellipse can be calculated using the center of the ellipse and c:
- The vertex on the positive x-axis is (h + c, k) = (2 + 4, -5) = (6, -5).
- The vertex on the negative x-axis is (h - c, k) = (2 - 4, -5) = (-2, -5).
Therefore, the center of the ellipse is (2, -5), and the vertices are (6, -5) and (-2, -5).