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Math  /  Geometry

QuestionQuestion
Which type of conic is represented by the equation below? 25x2−9y2−250x−36y+364=025 x^{2}-9 y^{2}-250 x-36 y+364=0
This is an equation of a hyperbola. Write the equation of this conic section in conic form.

Studdy Solution

STEP 1

1. The given equation is 25x2−9y2−250x−36y+364=0 25x^2 - 9y^2 - 250x - 36y + 364 = 0 .
2. The equation represents a hyperbola.
3. The goal is to rewrite the equation in the standard form of a hyperbola.

STEP 2

1. Rearrange the equation to group x x and y y terms.
2. Complete the square for both x x and y y terms.
3. Simplify and write the equation in standard form.

STEP 3

Rearrange the equation to group x x and y y terms:
25x2−250x−9y2−36y=−364 25x^2 - 250x - 9y^2 - 36y = -364

STEP 4

Complete the square for the x x terms:
1. Factor out 25 25 from the x x terms: 25(x2−10x) 25(x^2 - 10x)
2. Complete the square: x2−10x x^2 - 10x =(x−5)2−25 = (x - 5)^2 - 25
3. Substitute back: 25((x−5)2−25)=25(x−5)2−625 25((x - 5)^2 - 25) = 25(x - 5)^2 - 625

STEP 5

Complete the square for the y y terms:
1. Factor out −9 -9 from the y y terms: −9(y2+4y) -9(y^2 + 4y)
2. Complete the square: y2+4y y^2 + 4y =(y+2)2−4 = (y + 2)^2 - 4
3. Substitute back: −9((y+2)2−4)=−9(y+2)2+36 -9((y + 2)^2 - 4) = -9(y + 2)^2 + 36

STEP 6

Substitute completed squares back into the equation and simplify:
25(x−5)2−625−9(y+2)2+36=−364 25(x - 5)^2 - 625 - 9(y + 2)^2 + 36 = -364
Combine constants:
25(x−5)2−9(y+2)2−589=−364 25(x - 5)^2 - 9(y + 2)^2 - 589 = -364
Add 589 589 to both sides:
25(x−5)2−9(y+2)2=225 25(x - 5)^2 - 9(y + 2)^2 = 225
Divide the entire equation by 225 225 to get the standard form:
(x−5)29−(y+2)225=1 \frac{(x - 5)^2}{9} - \frac{(y + 2)^2}{25} = 1
The equation in conic form is:
(x−5)29−(y+2)225=1 \boxed{\frac{(x - 5)^2}{9} - \frac{(y + 2)^2}{25} = 1}

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