9x^2-4y^2-72x+8y+176=0

Question

Write its equation in standard form
Sketch the graph
For Parabolas, identify the vertex, focus, directrix, and axis of symmetry. For Circles, Identify the center and radius. For Ellipses, identify, the center, vertices, co-vertices, and foci. For Hyperbolas, identify the center, vertices, co-vertices, foci, and asymptotes

Answer 1

vertical hyperbola

rewrite the equation in the general form

(y-k)^2/b^2 - (x-h)^2/a^2 = 1 where (h,k) is the center point, midway between the two foci

center (4, 1/4), two branches are above and below the center

9x^2 -72x - 8y^2 + 4y + 172 = 0

8y^2 -4y - 9x^2 +72x = 172

complete the squares

8(y^2- y/2 + 1/16) - 9(x^2-8x +16) = 172+1/2- 144

8(y-1/2)^2 - 9(x-4)^2 = 28.5

(y-1/2)^2/(28.5/8) - [(x-4)^2]/3.2 = 1

b=sqr(28.5/8) a=sqr3.2

use the pythagorean theorem, c^2 =a^2+b^2

c=sqr(6.7625)= about 2.6

distance between the two foci is 2c

vertices are the center point, but with y coordinate plus and minus b

covertices are the center point, but with x coordinate plus and minus a

asymptotes: y-k = +/-(b/a)(x-h) where (h,k) is the hyperbola's center