Convert each general equation to standard form.
144X^2-25Y^2+864X-100Y-2404=0
Y^2-4X^2-12Y-16X+16=0
I will do the hard one, you do the easy one
144x^2 + 864x + .... - (25y^2 + 100y + ...) =2404
144(x^2 + 6x + ....) - 25(y^2 + 4y + ...) = 2404
144(x^2 + 6x + 9) - 25(y^2 + 4y + 4) = 2404 + 1296 - 100
144(x+3)2 - 25(y+2)^2 = 3600
divide each term by 3600
(x+3)^2 /25 - (y+2)^2 /144 = 1
do the 2nd one the same way
3455
To convert each general equation to standard form, we need to rearrange the terms to isolate the variables on one side and have the constant term on the other side.
Let's start with the first equation:
144X^2 - 25Y^2 + 864X - 100Y - 2404 = 0
To begin, let's regroup the terms by separating the X and Y terms:
(144X^2 + 864X) - (25Y^2 + 100Y) - 2404 = 0
Now, we can complete the square for both X and Y terms separately. Let's start with the X terms:
Taking half of the coefficient of X (which is 864) and squaring it:
(864/2)^2 = 36^2 = 1296
We add this value to both sides of the equation:
(144X^2 + 864X + 1296) - (25Y^2 + 100Y) - 2404 + 1296 = 1296
Now, let's complete the square for the Y terms:
Taking half of the coefficient of Y (which is 100) and squaring it:
(100/2)^2 = 50^2 = 2500
We add this value to both sides of the equation:
(144X^2 + 864X + 1296) - (25Y^2 + 100Y + 2500) - 2404 + 1296 = 1296 + 2500
Simplifying the equation:
144X^2 + 864X + 1296 - 25Y^2 - 100Y - 2500 - 2404 + 1296 = 1296 + 2500
Combining like terms:
144X^2 + 864X - 25Y^2 - 100Y - 3012 = 3796
Rearranging to get the standard form:
144X^2 - 25Y^2 + 864X - 100Y - 3012 = 3796
Therefore, the first equation in standard form is:
144X^2 - 25Y^2 + 864X - 100Y = 6808
Now, let's move on to the second equation:
Y^2 - 4X^2 - 12Y - 16X + 16 = 0
To begin, let's regroup the terms by separating the X and Y terms:
-4X^2 - 16X + Y^2 - 12Y = -16
Now, we can complete the square for both X and Y terms separately. Let's start with the X terms:
Taking half of the coefficient of X (which is -16) and squaring it:
(-16/2)^2 = (-8)^2 = 64
We add this value to both sides of the equation:
-4X^2 - 16X + 64 + Y^2 - 12Y = -16 + 64
Now, let's complete the square for the Y terms:
Taking half of the coefficient of Y (which is -12) and squaring it:
(-12/2)^2 = (-6)^2 = 36
We add this value to both sides of the equation:
-4X^2 - 16X + 64 + Y^2 - 12Y + 36 = -16 + 64 + 36
Simplifying the equation:
-4X^2 - 16X + 64 + Y^2 - 12Y + 36 = 84
Combining like terms:
-4X^2 - 16X + Y^2 - 12Y + 100 = 84
Rearranging to get the standard form:
-4X^2 - 16X + Y^2 - 12Y = -16
Therefore, the second equation in standard form is:
-4X^2 - 16X + Y^2 - 12Y = -16