Chapter 11, Problem 3RCC

a.

To determine

To find: The geometric definition of hyperbola and foci of hyperbola.

Answer to Problem 3RCC

Foci $=\left(\pm c,0\right)\to c^2=a^2+b^2$

Explanation of Solution

Given information:

The Geometric definition of hyperbola.

Concept Used:

Definition of Hyperbola:- It is a curve that is obtained when a plane intersects with both halves of double cone.

The foci are defined as the two fixed points that are located in each curve of hyperbola.

Conclusion:

This hyperbola is locus of points where the difference between the distance to each focus is constant.

b.

To determine

To find: The Foci, equation of asymptotes, transverse axis.

Answer to Problem 3RCC

Foci $=\left(\pm c,0\right)\to c^2=a^2+b^2$

Vertices $=\left(\pm a,0\right)$

Asymptotes, $y=\pm \frac{b}{a}x$

Transverse axis, $y=0$

Explanation of Solution

Given information:

Given equation: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Graph :

Given equation: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

  $\begin{array}{l}\text{Co-ordinates of vertices}=\left(a,0\right)or\left(-a,0\right)\\ \text{ Foci}=\left(c,0\right)or\left(-c,0\right)\\ \text{Here }c=\sqrt{a^2+b^2}\end{array}$

Equation of asymptotes are $y=\frac{b}{a}x\text{ and }y=\frac{-b}{a}x$

  

Interpretation :

Transverse axis is the line joining the two vertices of hyperbola. In this case it would be horizontal axis i.e., X-axis.

c.

To determine

To find: The equation of hyperbola with foci on y-axis.

Answer to Problem 3RCC

  $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\left(a^2>b^2\right)$

Explanation of Solution

Given information:

Foci is on y-axis.

Calculation:

If the foci is on the y-axis i.e. co-ordinates will be $\le \left(0,\pm c\right)$ with center origin. Transverse axis would be y-axis with length $2a$ .

  $\therefore \text{Vertices}=\left(\pm a,0\right)$

Let $2b$ be the length of the horizontal axis where $b<a$

  $\therefore c^2=a^2+b^2$

Therefore, the equation of the hyperbola will be

  $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

d.

To determine

To find: The steps to sketch a hyperbola with a given equation.

Answer to Problem 3RCC

Explanation of Solution

Given information:

Equation of hyperbola is

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Calculation:

With the given equation: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ , to sketch the graph, the variable $a\text{ or }b$ must be assigned with same numbers in such a way that $a>b>0$

  $\begin{array}{l}centre=\left(0,0\right)\\ foci=\left(\pm c,0\right)\text{ where }{c}^2=a^2+b^2\\ vertices=\left(\pm a,0\right)\end{array}$

Now for different values of $y$ obtain $x$ that form co-ordinates and plot the graph.

Therefore all the required co-ordinates from the hyperbola are obtained.