Problem 220 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The equation of the hyperbola whose foci are \((-2,0)\) and \((2,0)\) and eccentricity is 2 is given by : (a) \(x^{2}-3 y^{2}=3\) (b) \(3 x^{2}-y^{2}=3\) (c) \(-x^{2}+3 y^{2}=3\) (d) \(-3 x^{2}+y^{2}=3\)

Step-by-Step Solution

Answer

The equation is \(3x^2 - y^2 = 3\), which is option (b).

Step 1: Identify Key Elements of Hyperbola

The foci of the hyperbola are (-2,0) and (2,0), indicating that the hyperbola is centered at the origin with a horizontal transverse axis. The standard form for the equation of such a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The distance between the foci is 4, so 2c = 4\, thus\, c = 2.

Step 2: Utilize Eccentricity

The eccentricity \(e\) of a hyperbola is defined as \(e = \frac{c}{a}\). Here, \(e = 2\) and \(c = 2\). Thus, \(a = \frac{c}{e} = \frac{2}{2} = 1\). Therefore, \(a^2 = 1\).

Step 3: Relate \(b^2\) Using \(c^2 = a^2 + b^2\)

Use the relationship \(c^2 = a^2 + b^2\) to find \(b^2\). Substituting the known values, \(c^2 = 4\), \(a^2 = 1\), so \(b^2 = c^2 - a^2 = 4 - 1 = 3\).

Step 4: Form the Equation

Substitute \(a^2 = 1\) and \(b^2 = 3\) into the standard form equation for the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), yielding \(\frac{x^2}{1} - \frac{y^2}{3} = 1\) or \(x^2 - \frac{y^2}{3} = 1\).

Step 5: Rearrange to Match Answer Choices

Rearrange the equation \(x^2 - \frac{y^2}{3} = 1\) to match the form of the answer choices by multiplying through by 3: \(3x^2 - y^2 = 3\). This corresponds to option (b).

Key Concepts

Eccentricity

Eccentricity is a crucial concept when understanding hyperbolas. It measures how much the conic section (like a hyperbola) deviates from being circular.
The eccentricity of a hyperbola is defined as the ratio of the distance from any point on the hyperbola to its focus, to the perpendicular distance from that point to the nearest directrix.
In mathematical terms, if you have a hyperbola with foci located at a distance \(c\) from the center and a vertex distance \(a\) from the center, the eccentricity \(e\) is given by the formula \(e = \frac{c}{a}\).

• When \(e > 1\), it indicates a hyperbola.
• The greater the eccentricity, the more "stretched" the hyperbola looks.

For example, in this problem, the eccentricity is 2, implying the hyperbola is significantly stretched along its transverse axis compared to its conjugate axis.

Standard Form Equations

The equation of a hyperbola can often be found using its 'standard form.' This varies depending on whether the transverse axis is horizontal or vertical.
For a horizontal transverse axis, the typical standard form of the hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Conversely, for a vertical axis, it would be \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

• \(a^2\) represents the square of the distance from the center to each vertex on the transverse axis.
• \(b^2\) represents the square of the distance from the center to the vertices on the conjugate axis.

In the given solution, the problem finds the values for \(a^2\) and \(b^2\) using characteristics of the foci and eccentricity, thereby leading to accurately establishing the equation of the hyperbola.

Hyperbola Foci

The foci of a hyperbola are two fixed points located along the transverse axis. A hyperbola is defined as the set of all points such that the absolute difference of the distances to each focus is constant.
When you have the foci, you can determine many properties of the hyperbola.
For a hyperbola centered at the origin with a horizontal transverse axis, the foci are positioned at \((-c, 0)\) and \((c, 0)\), with \(c\) being the distance from the center to the focus.

• The distance \(2c\) between foci helps determine the major behavior of the hyperbola.
• For this problem, foci at \((-2, 0)\) and \((2, 0)\) suggest \(c = 2\).

By identifying the positions of the foci, solving this exercise becomes straightforward as you find values for other necessary parameters like \(a\) and \(b\), which lead to the exact form of the hyperbola's equation.