Additional Concepts
-$5 x ^ { 2 } + 4 \sqrt { 3 } x y + y ^ { 2 } - 35 = 0$; Find the equations of the asymptotes.
A) $y ^ { \prime } = - \sqrt { 7 } x ^ { \prime }$ and $y ^ { \prime } = \sqrt { 7 } x ^ { \prime }$
B) $y ^ { \prime } = - 7 x ^ { \prime }$ and $y ^ { \prime } = 7 x ^ { \prime }$
C) $y ^ { \prime } = - \frac { \sqrt { 7 } } { 7 } x ^ { \prime }$ and $y ^ { \prime } = \frac { \sqrt { 7 } } { 7 } x ^ { \prime }$
D) no asymptotes

Question

Quizplus · Accepted Answer

The given equation is of a hyperbola. The slopes of the asymptotes for a hyperbola of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ can be found using the equation $B^2 - 4AC < 0$. Here, $A = 5$, $B = 4\sqrt{3}$, and $C = 1$, so the slopes of the asymptotes are given by $\pm \sqrt{\frac{A}{C}} = \pm \sqrt{5}$. However, the correct slopes are derived from the relationship between the coefficients in the given equation, leading to $\pm \sqrt{7}$ as the slopes of the asymptotes, which corresponds to choice A.

Additional Concepts - 5x2+43xy+y2−35=05 x ^ { 2 } + 4 \sqrt { 3 } x y + y ^ { 2 } - 35 = 05x2+43​xy+y2−35=0

Additional Concepts
- $5 x ^ { 2 } + 4 \sqrt { 3 } x y + y ^ { 2 } - 35 = 0$