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

Answer:

the answer is d (x+4)^2/9 - (y-1)^2/25

Step-by-step explanation:

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The standard form of the given equation is [tex]\dfrac{(x+4)^2}{9}-\dfrac{(y+1)^2}{25}=1[/tex]

Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

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

25x² +200x - 9y² +18y = -166

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:

25(x² +8x) - 9 (y² +2y) = -166

Now take half the linear term in each set of parenthesis, square it, and add it 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:

25(x² +8x + 16)-9(y²+2y +1) = -166+400-9

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:

25(x+4)² - 9 (y+1)²= 225

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

[tex]\dfrac{(x+4)^2}{9}-\dfrac{(y+1)^2}{25}=1[/tex]

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.

Therefore the standard form of the given equation is[tex]\dfrac{(x+4)^2}{9}-\dfrac{(y+1)^2}{25}=1[/tex]

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