Problem 44 - A Complete Resource Book in Mathematics for JEE Main

Question

The slopes of common tangents to the hyperbolas \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\) are (A) \(\pm 2\) (B) \(\pm 1\) (C) \(\pm \sqrt{2}\) (D) none of these

Step-by-Step Solution

Answer

The slopes of the common tangents are \( \pm \sqrt{2} \).

Step 1: Identify the General Equation of the Tangent to a Hyperbola

For the given hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \), the equation of the tangent line can be expressed by the equation \( \frac{xx_1}{9} - \frac{yy_1}{16} = 1 \). Similarly, for the hyperbola \( \frac{y^2}{9} - \frac{x^2}{16} = 1 \), the tangent line is \( \frac{yy_1}{9} - \frac{xx_1}{16} = 1 \).

Step 2: Set Up the Equations for Common Tangents

For the lines that are common tangents, they should satisfy both hyperbolas simultaneously. Since we are looking for the slope \(m\) of these common tangents, assume a line of the form \( y = mx + c \). This equation should represent tangents to both hyperbolas.

Step 3: Use the Equation Form of Hyperbolic Tangents

Substitute \( y = mx + c \) into the tangent equations obtained from both hyperbolas. For hyperbola 1: \( \frac{x \, x_1}{9} - \frac{(mx + c) \, y_1}{16} = 1 \) and for hyperbola 2: \( \frac{(mx + c) \, y_1}{9} - \frac{x \, x_1}{16} = 1 \). Solve for the slope \(m\) from these equations.

Step 4: Simplify and Solve for Slope

By comparison and rearrangement of the tangent equations, the condition for a common tangent is given by solving the determinant condition formed by the system of linear equations derived in Step 3. This gives a pair of solutions for \(m\).

Step 5: Calculate Specific Slope Values

After simplifying from the determinant, derive the possible slope values. Using the property of the determinant derived previously, this reduces to calculating roots of a specific quadratic equation \( m^2 - 2 = 0 \). Solve this for the possible values of \(m\).

Step 6: Conclude the Slope Values

From the quadratic equation \( m^2 - 2 = 0 \), you obtain \( m = \pm \sqrt{2} \). Compare these results to the given answer choices.

Key Concepts

hyperbola

A hyperbola is a type of conic section that appears as an open curve. Unlike ellipses, hyperbolas have two disconnected curves called branches. These branches are mirror images of each other. In mathematics, hyperbolas are described by specific equations that show how the points are distributed in relation to two fixed points known as foci.

The standard form of a hyperbola's equation depends on its orientation:

• For a horizontal hyperbola, the equation is \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\.
• For a vertical hyperbola, it is \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\.

The constants \(a\) and \(b\) determine the hyperbola’s shape and size, with \(a\) as the distance from the center to a vertex along the transverse axis.

Hyperbolas are vital in mathematics due to their geometric properties and ability to model phenomena such as the paths of space objects and certain sound waves.

slope determination

Determining the slope of a line is crucial in understanding its angle of inclination with the horizontal axis. The slope, often denoted by \(m\), is a measure of the steepness or direction of a line, with the formula \( m = \frac{\Delta y}{\Delta x} \).

When dealing with equations like that of a common tangent to two hyperbolas, understanding the slope allows you to express the line equation as \( y = mx + c \). The challenge is to ensure this line can be a tangent to both hyperbolas simultaneously.

To find the slope of common tangents, it often involves comparing equations of the tangents derived from the hyperbola's general equation. You may need to solve a quadratic equation which might arise from equating two derivations, leading to the possible values of \(m\). This ensures the slope meets the conditions required to be a tangent to both specified hyperbolas.

equation of tangent

The equation of a tangent line to a hyperbola is derived based on its standard equation. The tangent line’s equation to a hyperbola of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) can generally be expressed as \( \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \). This equation helps ascertain which line touches the hyperbola at only one point without crossing it.

When finding common tangents, these equations serve as a basis. The form \( y = mx + c \) should satisfy both hyperbolic tangent conditions. By substituting \( y = mx + c \) into each hyperbola's tangent equation, constraints can be set to find possible \(c\) for a given \(m\), thus determining if real tangents exist for that slope. Ultimately, only specific slope values enable a tangent to be common to both hyperbolas.

JEE mathematics

The Joint Entrance Examination (JEE) is an engineering entrance exam held annually in India, which observes rigorous standards. In the mathematics section of JEE, concepts such as hyperbolas and their properties, like common tangents, frequently appear and are critical to solving complex problems quickly.

Mastering topics such as the identification of hyperbolas, understanding the derivation of their equations, and solving problems involving common tangents often grants a substantial advantage in achieving a high score. Thus, learning how to determine the slopes suitable for common tangents and applying these skills in both theoretical and practical problems are essential for JEE candidates. This involves a deep understanding of the related algebra, geometry, and sometimes trigonometry, with practice problems often incorporating multiple steps.