Change to standard form 2x^2+4y^2-12x-64y-16=0 i don't know how to solve it.
you will have to "complete the square"
2x^2+4y^2-12x-64y-16=0
2(x^2 - 6x + ....) + 4(y^2 - 16y + ....) = 16
2(x^2 - 6x + 9) + 4(y^2 - 16y + 64) = 16 + 2(9) + 4(64)
2(x-3)^2 + 4(y-8)^2 = 290
divide each term by 290
2(x-3)^2 /290 + 4(y-8)^2 / 290 = 1
(x-3)^2 / 145 + (y-8)^2 / (145/2) = 1
standard form:
(x-h)^2 /a^2 + (y-k)^2 /b^2 = 1 <----- ellipse with major axis as 2a, and minor axis as 2b, centre (h,k)
so centre is (3,8)
a = √145 , b = √(145/2)
2 x ^ 2 + 4 y ^ 2 - 12 x - 64 y - 16 = 0 Add 16 to both sides
2 x ^ 2 + 4 y ^ 2 - 12 x - 64 y - 16 + 16 = 0 + 16
2 x ^ 2 + 4 y ^ 2 - 12 x - 64 y = 16
2 x ^ 2 - 12 x + 4 y ^ 2 - 64 y = 16
2 ( x ^ 2 - 6 x ) + 4 ( y ^ 2 - 16 y ) = 16 Divide both sides by 4
2 ( x ^ 2 - 6 x ) / 4 + 4 ( y ^ 2 - 16 y ) / 4 = 16 / 4
( 2 / 4 ) ( x ^ 2 - 6 x ) + ( 4 / 4 ) ( y ^ 2 - 16 y ) = 4
( 1 / 2 ) ( x ^ 2 - 6 x ) + ( y ^ 2 - 16 y ) = 4 Add 9 / 2 to both sides
( 1 / 2 ) ( x ^ 2 - 6 x ) + ( y ^ 2 - 16 y ) + 9 / 2 = 4 + 9 / 2
( 1 / 2 ) ( x ^ 2 - 6 x ) + 9 / 2 + ( y ^ 2 - 16 y ) = 8 / 2 + 9 / 2
( 1 / 2 ) ( x ^ 2 - 6 x + 9 ) + ( y ^ 2 - 16 y ) = 17 / 2 Add 64 to both sides
( 1 / 2 ) ( x ^ 2 - 6 x + 9 ) + ( y ^ 2 - 16 y ) + 64 = 17 / 2 + 64
( 1 / 2 ) ( x ^ 2 - 6 x + 9 ) + ( y ^ 2 - 16 y + 64 ) = 17 / 2 + 128 / 2
( 1 / 2 ) ( x ^ 2 - 6 x + 9 ) + ( y ^ 2 - 16 y + 64 ) = 145 / 2
Since the ( x ^ 2 - 6 x + 9 ) = ( x - 3 ) ^ 2 and y ^ 2 - 16 y + 64 = ( y - 8 ) ^ 2
( 1 / 2 ) ( x - 3 ) ^ 2 + ( y - 8 ) ^ 2 = 145 / 2 Multiply both sides by 2
( 1 / 2 ) * 2 * ( x - 3 ) ^ 2 + 2 * ( y - 8 ) ^ 2 = 145 * 2 / 2
( x - 3 ) ^ 2 + 2 ( y - 8 ) ^ 2 = 145 Divide both sides by 145
( x - 3 ) ^ 2 / 145 + 2 ( y - 8 ) ^ 2 / 145 = 145 / 145
( 1 / 145 ) ( x - 3 ) ^ 2 + ( 2 / 145 ) ( y - 8 ) ^ 2 = 1
To change the given equation into standard form, which is written in the form of Ax^2 + By^2 + Cx + Dy + E = 0, you need to follow these steps:
Step 1: Rearrange the equation. Move all the constant terms to the right side of the equation.
2x^2 + 4y^2 - 12x - 64y - 16 = 0
Rearranging the terms gives:
2x^2 - 12x + 4y^2 - 64y = 16
Step 2: Complete the square for x terms and y terms separately.
For the x terms:
- Take the coefficient of x (A) and divide it by 2, and then square that value.
- Add that result to both sides of the equation.
In this case, the coefficient of x (A) is 2.
(2x^2 - 12x) + __ = 16
- Take half of -12 (which is -6) and square (-6)^2 = 36.
- Add 36 to both sides of the equation.
(2x^2 - 12x + 36) + __ = 16 + 36
Now, rewrite the x terms as a perfect square.
(2x^2 - 12x + 36) + 4y^2 - 64y = 52
For the y terms:
- Take the coefficient of y (B) and divide it by 2, and then square that value.
- Add that result to both sides of the equation.
In this case, the coefficient of y (B) is 4.
(4y^2 - 64y) + __ = 52
- Take half of -64 (which is -32) and square (-32)^2 = 1024.
- Add 1024 to both sides of the equation.
(4y^2 - 64y + 1024) + (2x^2 - 12x + 36) = 52 + 1024
Now, rewrite the y terms as a perfect square.
(2x^2 - 12x + 36) + (4y^2 - 64y + 1024) = 1076
Step 3: Simplify the equation.
(2x - 6)^2 + (2y - 32)^2 = 1076
The equation is now in standard form:
(2x - 6)^2 + (2y - 32)^2 = 1076
This equation represents an ellipse centered at the point (3, 16) with major and minor axes determined by the square roots of 1076/2.