Hyperbola quiz Grade 12

📚 Quick Study Guide

🔎 A hyperbola is defined as the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant.
• 💡 The standard form of a hyperbola centered at $(h, k)$ with a horizontal transverse axis is $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$. The vertices are $(h \pm a, k)$, and the foci are $(h \pm c, k)$, where $c^2 = a^2 + b^2$.
• 📝 The standard form of a hyperbola centered at $(h, k)$ with a vertical transverse axis is $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$. The vertices are $(h, k \pm a)$, and the foci are $(h, k \pm c)$, where $c^2 = a^2 + b^2$.
• 📐 The asymptotes of a hyperbola $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ are given by $y - k = \pm \frac{b}{a}(x - h)$.
• 📈 The asymptotes of a hyperbola $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ are given by $y - k = \pm \frac{a}{b}(x - h)$.

Practice Quiz

1. What is the standard form equation of a hyperbola centered at (0,0) with a horizontal transverse axis?
   1. $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
   2. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
   3. $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
2. The distance between the foci of a hyperbola is 10, and the distance between the vertices is 8. What is the value of 'c'?
   1. 4
   2. 5
   3. 6
3. What is the equation of the asymptote for the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$?
   1. $y = \pm \frac{4}{3}x$
   2. $y = \pm \frac{3}{4}x$
   3. $y = \pm \frac{9}{16}x$
4. The hyperbola $\frac{(y-2)^2}{4} - \frac{(x-1)^2}{9} = 1$ is centered at which point?
   1. (2, 1)
   2. (1, 2)
   3. (-1, -2)
5. What is the value of $c^2$ for the hyperbola $\frac{x^2}{25} - \frac{y^2}{144} = 1$?
   1. 119
   2. 169
   3. 13
6. Which of the following describes the orientation of the transverse axis of the hyperbola $\frac{(y+1)^2}{16} - \frac{(x-3)^2}{9} = 1$?
   1. Horizontal
   2. Vertical
   3. Oblique
7. What is the value of 'a' for the hyperbola described by the equation $\frac{(x-2)^2}{16} - \frac{(y+3)^2}{9} = 1$?
   1. 3
   2. 4
   3. 5