Convert the equation into standard form 25x^2 -9y^2+200x +18y +166=0

Step-by-step explanation:

You'd have to be really familiar with conic sections to know what to do with this. Good thing I am! ; )

Begin by grouping the x terms together and the y terms together, and getting the constant on the other side of the equals sign:

Now we need to complete the square on the x terms and the y terms. Do this by first factoring out the leading coefficient from each, the 25 from the x's and the 9 from the y's:

Now take half the linear term in each set of parenthesis, square it, and add it in to both sides, remembering the multiplier outside (the 25 and the 9). Our x linear term is 8. Half of 8 is 4, and 4 squared is 16, so we add a 16 into the parenthesis with the x's; our y linear term is 2. Half of 2 is 1 and 1 squared is 1, so we add a 1 into the parenthesis with the y's:

Note the 400 and -9 on the right side now. 25 times 16 is 400; we didn't just add in a 16, we have to multiply the scalar number into it before we know what we REALLY added in. And the -9 comes from multiplying the -9 times 1.

The reason we do this is to get the perfect square binomials on the left that we created while completing the square:

Now, finally we will divide both sides by 225 to get this conic into standard form:

This is a hyperbola with a horizontal transverse axis and a center of (-4, -1). The reason we know it's not an ellipse is because an ellipse will always have a + sign separating the x-squared from the y-squared whereas a hyperbola always has a - sign separating them. And we also know it's not a circle because the values of the leading coefficients weren't the same.

the answer is true.

step-by-step explanation:

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: