Question

Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a>b>0$, be a hyperbola in the $xy$ -plane whose conjugate axis $LM$ subtends an angle of ${60}^{\circ }$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4\sqrt{3}$ LIST-I P. The length of the conjugate axis of $H$ is Q. The eccentricity of $H$ is $\mathbf{R}$. The distance between the foci of $H$ is S. The length of the latus rectum of $H$ is LIST-II 1\. 8 2\. $\frac{4}{\sqrt{3}}$ 3\. $\frac{2}{\sqrt{3}}$ 4\. 4

Short answer

P. The length of the conjugate axis of $H$ is 4. Q. The eccentricity of $H$ is $\frac{2}{\sqrt{3}}$. R. The distance between the foci of $H$ is 8. S. The length of the latus rectum of $H$ is $\frac{4}{\sqrt{3}}$.

Step-by-step solution

Determining the Length of the Conjugate Axis

We are given the area of triangle LMN, formed by the conjugate axis LM and vertex N, is $4\sqrt{3}$. Since the angle at N is ${60}^{\circ }$, we use the formula for the area of a triangle ($\text{Area}=\frac{1}{2}ab\sin (\gamma )$) to find the length of LM (conjugate axis), which is $2b$. So $\frac{1}{2}(2a)(2b)\sin ({60}^{\circ })=4\sqrt{3}$, solving this will give us the value of $b$.

Calculating the Length of the Conjugate Axis

We calculate the length of the conjugate axis by substituting the known values into the area equation: $2ab\sin ({60}^{\circ })=8\sqrt{3}$. Solving, we get $ab\sin ({60}^{\circ })=4\sqrt{3}$, and since $\sin ({60}^{\circ })=\frac{\sqrt{3}}{2}$, the equation simplifies to $ab\sqrt{3}/2=4\sqrt{3}$ which leads to $ab=8$. As $a>b$, and we need to find the length of the conjugate axis ($2b$), we get that $2b=4$.

Finding the Eccentricity of the Hyperbola

The eccentricity $e$ of a hyperbola is given by the relationship $e^2=1+\frac{b^2}{a^2}$. Now that we have $b$, we need to find $a$ using the relation $ab=8$ we obtained earlier. Once $a$ is known, we can solve for the eccentricity $e$.

Calculating the Distance Between the Foci

The distance between the two foci of the hyperbola is given by $2ae$ where $e$ is the eccentricity. With $a$ and $e$ determined from previous steps, we can readily calculate this distance.

Determining the Length of the Latus Rectum

The latus rectum of a hyperbola is given by the length $\frac{2b^2}{a}$. Since we have $b$ and will have calculated $a$, we can substitute these values to find the length of the latus rectum.

Key concepts

Conjugate Axis of a Hyperbola

In the fascinating world of hyperbola geometry, the conjugate axis plays a pivotal role. It's perpendicular to the transverse axis and passes through the center of the hyperbola. Think of it as the 'lesser known twin' of the more prominent transverse axis associated with the hyperbola's open ends. For a hyperbola represented by the equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the length of the conjugate axis is twice the value of 'b'.

As an example, consider a scenario where we know the area of a triangle formed with the hyperbola’s conjugate axis and one of its vertices. Using this area, along with a given angle that the axis subtends at the vertex, we can deduce the length of the conjugate axis. This understanding is crucial because the conjugate axis provides insights into the hyperbola's dimensional spread perpendicular to the direction of opening.

Eccentricity of a Hyperbola

The eccentricity of a hyperbola, usually denoted by 'e', measures how 'stretched' the hyperbola is. Unlike circles which have an eccentricity of zero, hyperbolas have an eccentricity greater than one. This peculiar measure, given by the equation $e^2=1+\frac{b^2}{a^2}$ for a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, captures the deviation from a perfect circle to the elongated shape of our hyperbola.

The eccentricity is literally the 'eccentric' character of hyperbolas that sets them apart in the conic section family. It is a concept vital to understanding the nature of conic sections, and it comes into play in various applications such as satellite orbit calculations and in the study of planetary motion.

Distance Between Foci

Diving deeper into hyperbola geometry, the distance between foci is a concept that encapsulates the essence of the hyperbola's stretched form. For the standard-form equation of a hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, the foci (plural of 'focus') are two fixed points located along the transverse axis on either side of the center. The distance between these foci is critical for understanding how far the foci are spread out and is represented by $2ae$, where 'a' is the length of the semi-transverse axis and 'e' is the eccentricity of the hyperbola.

This measure is a key characteristic when analyzing the shape and properties of the hyperbola. It also plays an essential role when applying hyperbolic functions to real-world problems, such as in the fields of acoustics and optics where hyperbolic mirrors are used.

Latus Rectum of a Hyperbola

Last but certainly not least, the latus rectum of a hyperbola might sound like an obscure term, but it's actually quite a fundamental component in understanding this conic section. It refers to the line segment perpendicular to the transverse axis that passes through either focus and whose endpoints lie on the hyperbola. The length of the latus rectum plays a significant part in defining the conic's curvature and is given by the formula $\frac{2b^2}{a}$ for the equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

Understanding the latus rectum is not only important in pure geometry but also has practical implications in designing objects like satellite dishes and telescopes, which often involve hyperbolically curved surfaces to properly focus light or radio waves.