Graph the inequality . Write the coordinates for the vertices and find the equations for the asymptotes.

Question

y^2 /81 - x^2/ 49 > 1

Answer 1

Graphing of inequality is y ²/81 - x²/49 > 1 :

• First graph the hyperbola : y ²/81 - x²/49 = 1.
• The hyprerbola divides the plane into three regions.
• Select a test point in each region, and check to see wheather it satisfies the inequality.
• Testing the point (0, 12), (0, 0), and (0, - 12).

(12) ²/81 - (0)²/49 > 1 , (0) ²/81 - (0)²/49 > 1, and (- 12) ²/81 - (0)²/49 > 1

1.77 >0, 0 >1, and 1.77 > 0.

• Because the first and third are correct, shade only the regions contains (0, 12) and (0, - 12).

Graph :

• 

The hyperbola is y ²/81 - x²/49 = 1.

The standard form of the equation of a hyperbola with center at the origin (where a and b are not equals to 0) is x²/a² - y²/b² = 1 (Transverse axis is horizontal) or y²/a² - x²/b² = 1 (Transverse axis is vertical).

The vertices and foci are, respectively a and c units from the center and the relation between a, b and c is b² = c² - a².

Compare the equation y²/81 - x²/49 = 1 with y²/a² - x²/b² = 1.

a² = 81 and b² = 49.

a = ± 9 and b = ± 7.

To find the value of c, substitute the value of a² = 81 and b² = 49 in b² = c² - a².

49 = c² - 81

c² = 130

c = ± √130.

Here the transverse axis is vertical, the asymptotes are of the forms y = (a/b) x and y = - (a/b) x.

The asymptote equation is y = ± (9/7)x.

The  Vertices are = (0, ± a) = (0, ± 9) and asymptotes are y = ± (9/7)x.