Identify the points and graph the equation of the ellipse. 9x^2 + 4y^2 -54x-16y + 61 =0
what do you mean "identify the points"?
9x^2 + 4y^2 -54x-16y + 61 =0
9x^2-54x + 4y^2-16y = -61
9(x^2-6x) + 4(y^2-4y) = -61
9(x^2-6x+9) + 4(y^2-4y+4) = -61 + 9*9 + 4*4
9(x-3)^2 + 4(y-2)^2 = 36
(x-3)^2/4 + (y-2)^2/9 = 1
ellipse with major axis vertical, semi-axes of length 2 and 3.
Center at (3,2)
Foci at (3,2±√5)
e = √5/3
http://www.wolframalpha.com/input/?i=foci+of+9x^2+%2B+4y^2+-54x-16y+%2B+61+%3D0+
To identify the points and graph the equation of the ellipse 9x^2 + 4y^2 - 54x - 16y + 61 = 0, we need to rearrange the equation into standard form and identify the key components. The standard form equation of an ellipse is:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
Where (h, k) represents the center of the ellipse, "a" is the length of the major axis (horizontal), and "b" is the length of the minor axis (vertical).
Let's start by rearranging the given equation:
9x^2 + 4y^2 - 54x - 16y + 61 = 0
First, we can group the x-terms and y-terms together:
(9x^2 - 54x) + (4y^2 - 16y) + 61 = 0
Next, we complete the square for both the x-terms and y-terms separately:
(9(x^2 - 6x)) + (4(y^2 - 4y)) + 61 = 0
To complete the square inside the parentheses, we take half of the coefficient of the linear term, square it, and add it inside the parentheses. For the x-terms:
(9(x^2 - 6x + 9)) + (4(y^2 - 4y)) + 61 = 0
Similarly, for the y-terms:
(9(x^2 - 6x + 9)) + (4(y^2 - 4y + 4)) + 61 = 0
Now, we can simplify further:
(9(x - 3)^2) + (4(y - 2)^2) + 61 = 0
To isolate the equation, we subtract 61 from both sides:
(9(x - 3)^2) + (4(y - 2)^2) = -61
Now, we divide both sides of the equation by -61:
[(9(x - 3)^2) / -61] + [(4(y - 2)^2) / -61] = 1
Finally, we rearrange the equation into standard form:
[(x - 3)^2 / (9 / -61)] + [(y - 2)^2 / (4 / -61)] = 1
From the rearranged equation, we can identify the center, semi-major axis, and semi-minor axis:
Center: (3, 2)
Semi-Major Axis: √(-61/9)
Semi-Minor Axis: √(-61/4)
To graph the ellipse, start by plotting the center point (3, 2). Then, draw the major axis horizontally with a length of 2 times the semi-major axis. Draw the minor axis vertically with a length of 2 times the semi-minor axis.
Please note that the given equation contains negative values inside the square roots, which indicates an imaginary ellipse that cannot be graphed on a standard Cartesian plane.