Identify the type of conic section represented by 9x^2 - 25y^2 - 18x + 50y = 0?

Answer 1

Completing the square will help you with this problem:

1. Group all the x's and y's together:

9x^2 - 18x - 25y^2 + 50y = 0

2. Factor so that the coefficients in front of the x^2 and y^2 terms are 1:

9(x^2 - 2x) - 25(y^2 - 2y) = 0

3. Prepare for completing the square:

9(x^2 - 2x + ___) - 25(y^2 - 2y + ____) = 9(____) - 25(____)

Notice that I put blanks on both sides to keep the equation balanced.

4. Complete the square:

To find the numbers that would go in the blanks, you want to take the middle coefficient in each expression, divide it by two and square it.

middle coefficient is (-2) for both cases => (-2)^2/2 = 1

9(x^2 - 2x + 1) - 25(y^2 - 2y + 1) = 9(1) - 25(1)

5. Simplify

Now you have perfect squares on the left side!
9(x-1)^2 - 25(y-1)^2 = -16
25(y-1)^2 - 9(x-1)^2 = 16
(25/16)(y-1)^2 - (9/16)(x-1)^2 = 1

6. Identify the conic

Since you are subtracting the two squares, the only possible conic is a hyperbola.

More information: This hyperbola is centered at (1,1) and opens up and down (since y comes first). You can also determine the asymptotes of this hyperbola by looking at the coefficients.

Answer 2

a) sparkling up by technique of utilising factorising x2 - 6x + 8 = 0 smash up the 8 into its components, and then become attentive to 2 which upload to grant -6 and multiply to grant 8. you will get -2 and -4. subsequently, (x-2)(x-4) = 0 --- ab = 0, a=0 or b=0 x= 2 or 4. --------------------------- b) sparkling up utilising the quadratic formula. x= [-b±?(b2 - 4ac)] / 2a in x2 - 6x + 8 = 0, a = a million, b = -6 and c = 8 x = [6±?(36-32)] / 2 x = [6±2]/2 x = 2 or 4 --------------------------- 2) For the function y = x2 - 6x + 8, carry out right here initiatives: a) located the function interior the type y = a(x - h)2 + ok. it is composed of winding up the sq.. y = x2 - 6x + 8 y = x2 - 6x + 9 - a million y = (x-3)2-a million ---------------------------- b) what's the equation for the line of symmetry for the graph of this function? From section (a), the equation for the line of symmetry is x=h, subsequently x = 3 ---------------------------- c) [Graph] it extremely isn't had to plan components, as you will locate the equation of the line of symmetry, the vertex or turning element and intercepts very extremely. ---------------------------- d) The graph has an identical shape, yet is shifted a million unit down, and 3 instruments to the purely suited. ---------------------------- 3) you're given the purely suited equation and each and each and each and all of the variables, so purely pop them in and you get: s = -16t2 + 32t --------------------------- b) Sub in t=a million, s = -sixteen(a million)2+32(a million) = 16ft. --------------------------- c) mutually because it hits the floor, the s = 0. 0 = -16t2 + 32t 0 = t2 - 2t 0 = t(t-2) t = 0 or 2 seconds. Discarding the 0 answer, we get t = 2 seconds. ---------------------------- d) the utmost top is halfway between launch and hitting the floor, at t = a million 2d. you ought to use the respond from b, 16ft. ----------------------------------- 4) The equation for the fringe is: 2(l+w) = 400ft. l + w = 200ft. l = 2 hundred - w. The equation for the section is A = l x w. sub in (2 hundred-w) for l, A = w(2 hundred-w) A = - w2 + 200w we could consistently detect the turning (optimal) element. The equation for the line of symmetry is -b/2a = -2 hundred/-2 = a hundred subsequently on the max, w = 100ft and l = 2 hundred-a hundred = 100ft. the section it extremely is 10,000 ft2.

Answer 3

This is a hyperbola because the x² and y² coefficients
have different signs.