Find the asymptotes of the hyperbola.
-$$x ^ { 2 } - y ^ { 2 } - 6 x + 8 y - 11 = 0$$

A) $y - 4 = ( x - 3 )$ and $y - 4 = - ( x - 3 )$

B) $y + 3 = ( x + 4 )$ and $y + 3 = - ( x + 4 )$

C) $y - 3 = ( x - 4 )$ and $y - 3 = - ( x - 4 )$

D) $y - 4 = \frac { 1 } { 2 } ( x - 3 )$ and $y - 4 = - \frac { 1 } { 2 } ( x - 3 )$

Question

Find the asymptotes of the hyperbola.
-$$x ^ { 2 } - y ^ { 2 } - 6 x + 8 y - 11 = 0$$

A) $y - 4 = ( x - 3 )$ and $y - 4 = - ( x - 3 )$

B) $y + 3 = ( x + 4 )$ and $y + 3 = - ( x + 4 )$

C) $y - 3 = ( x - 4 )$ and $y - 3 = - ( x - 4 )$

D) $y - 4 = \frac { 1 } { 2 } ( x - 3 )$ and $y - 4 = - \frac { 1 } { 2 } ( x - 3 )$

Quizplus · Accepted Answer

The asymptotes of a hyperbola are found by rewriting the equation in its standard form and then using the coefficients of $x^2$ and $y^2$ to determine the slopes of the asymptotes. For the given equation, completing the squares for $x$ and $y$ terms gives us the standard form of the hyperbola: $$(x-3)^2 - (y-4)^2 = 16.$$ The asymptotes for this standard form are given by the equations $y - 4 = \pm(x - 3)$, which matches choice A.

Find the Asymptotes of the Hyperbola x2−y2−6x+8y−11=0x ^ { 2 } - y ^ { 2 } - 6 x + 8 y - 11 = 0x2−y2−6x+8y−11=0

Find the Asymptotes of the Hyperbola $x ^ { 2 } - y ^ { 2 } - 6 x + 8 y - 11 = 0$