Classify the conic and write the equation in standard form:
(1) 9x^2 + 9y^2 + 18x - 18y = 0
(2) 16x^2 + 16y^2 - 64x + 32y + 55 = 0
(3) 6x^2 + 2y^2 + 18x - 10y + 2 = 0
the first two are circles, the last one is an ellipse
I will do the second one by
completing the square,
16x^2 + 16y^2 - 64x + 32y + 55 = 0
16(x^2 - 4x+ ___) + 16(y^2 + 2y + ___) = -55
16(x^2 - 4x+ 4) + 16(y^2 + 2y + 1) = -55+64+16
16(x^2 - 4x+ 4) + 16(y^2 + 2y + 1) = 25
(x-2)^2 + (y+1)^2 = 25/16
Oh, I see what I was doing wrong. I thought you had to SUBTRACT the numbers from the other side when you complete the square, but you have to ADD them. I get it now. Thanks!
To classify the conic and write the equation in standard form, follow these steps:
Step 1: Rearrange the equation by completing the square.
Step 2: Simplify the equation to determine the standard form.
Step 3: Identify the conic by examining the coefficients of the x^2 and y^2 terms.
Let's solve each equation one by one:
(1) 9x^2 + 9y^2 + 18x - 18y = 0
Step 1: Rearrange the equation by completing the square.
To complete the square for the x terms, we add and subtract (18/2)^2 = 81.
Similarly, for the y terms, we add and subtract (18/2)^2 = 81.
9x^2 + 18x + 9y^2 - 18y = 81
Step 2: Simplify the equation.
Factor out common terms from both the x and y groups.
9(x^2 + 2x) + 9(y^2 - 2y) = 81
Step 3: Identify the conic.
Since the coefficients of both the x^2 and y^2 terms are positive and equal, this equation represents a circle.
The standard form equation for the circle is given by ((x - h)^2/a^2) + ((y - k)^2/b^2) = r^2, where (h, k) is the center of the circle, a and b are the lengths of the major and minor axes, respectively, and r is the radius.
By comparing the given equation with the standard form, we can conclude that:
Center of the circle (h, k): (-1, 1)
Radius of the circle (r): 3
Therefore, the equation in standard form is:
((x + 1)^2/3^2) + ((y - 1)^2/3^2) = 1
(2) 16x^2 + 16y^2 - 64x + 32y + 55 = 0
Step 1: Rearrange the equation by completing the square.
To complete the square for the x terms, we add and subtract (64/2)^2 = 256.
For the y terms, add and subtract (32/2)^2 = 256.
16x^2 - 64x + 16y^2 + 32y = -55 + 256 + 256
Step 2: Simplify the equation.
Factor out common terms from both the x and y groups.
16(x^2 - 4x) + 16(y^2 + 2y) = 457
Step 3: Identify the conic.
The coefficients of both the x^2 and y^2 terms are positive but not equal. Additionally, the coefficients of the x and y terms are not equal. Hence, this equation represents an ellipse.
The standard form equation for an ellipse with a horizontal major axis is ((x - h)^2/a^2) + ((y - k)^2/b^2) = 1, where a is the length of the semi-major axis and b is the length of the semi-minor axis.
By comparing the given equation with the standard form, we can conclude that:
Center of the ellipse (h, k): (2, -1)
Length of semi-major axis (a): √(457/16)
Length of semi-minor axis (b): √(457/16)
Therefore, the equation in standard form is:
((x - 2)^2/√(457/16))^2 + ((y + 1)^2/√(457/16))^2 = 1
(3) 6x^2 + 2y^2 + 18x - 10y + 2 = 0
Step 1: Rearrange the equation by completing the square.
To complete the square for the x terms, we add and subtract (18/2)^2 = 81/4.
For the y terms, add and subtract (-10/2)^2 = 25.
6x^2 + 18x + 2y^2 - 10y = -2 + 81/4 + 25
Step 2: Simplify the equation.
Factor out common terms from both the x and y groups.
6(x^2 + 3x) + 2(y^2 - 5y) = -8 + 81/4
Step 3: Identify the conic.
The coefficients of both the x^2 and y^2 terms have the same sign but are not equal. Additionally, the coefficients of the x and y terms are not equal. Hence, this equation represents an ellipse.
The standard form equation for an ellipse with a vertical major axis is ((x - h)^2/b^2) + ((y - k)^2/a^2) = 1, where a is the length of the semi-major axis and b is the length of the semi-minor axis.
By comparing the given equation with the standard form, we can conclude that:
Center of the ellipse (h, k): (-3/2, 5/2)
Length of semi-major axis (a): √(81/24)
Length of semi-minor axis (b): √(81/6)
Therefore, the equation in standard form is:
((x + 3/2)^2/√(81/24))^2 + ((y - 5/2)^2/√(81/6))^2 = 1