Question

The equations to the common tangents to the two hyperbola \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\) and are \(\left(y^{2} / a^{2}\right)-\left(x^{2} / b^{2}\right)=1\) (a) \(\mathrm{y}=\pm \mathrm{x} \pm \sqrt{\left(\mathrm{a}^{2}-\mathrm{b}^{2}\right)}\) (b) \(\mathrm{y}=\pm \mathrm{x} \pm \sqrt{\left(\mathrm{b}^{2}-\mathrm{a}^{2}\right)}\) (c) \(\mathrm{y}=\pm \mathrm{x} \pm \sqrt{\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)}\) (d) \(\mathrm{y}=\pm \mathrm{x} \pm\left(\mathrm{a}^{2}-\mathrm{b}^{2}\right)\)

Short answer

\(y = \pm x \pm \sqrt{a^2 + b^2}\)

Step-by-step solution

Identify the given hyperbolas

The given hyperbolas are: Hyperbola 1: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) Hyperbola 2: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

Find the general equation of a tangent line to Hyperbola 1

A general equation of a tangent line to Hyperbola 1 can be obtained by replacing \(x^2\) by \(x x_1\) and \(y^2\) by \(y y_1\), where \((x_1, y_1)\) is a point on the tangent line. We have: Tangent to Hyperbola 1: \(\frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1\)

Find the general equation of a tangent line to Hyperbola 2

Similarly, we can obtain the general equation of a tangent line to Hyperbola 2: Tangent to Hyperbola 2: \(\frac{y y_1}{a^2} - \frac{x x_1}{b^2} = 1\)

Find the relationship between the slopes of the tangents

The slopes of two perpendicular lines multiply to -1. So, we have: Slope of Tangent to Hyperbola 1 × Slope of Tangent to Hyperbola 2 = -1

Find the required common tangents from the given equations

Now we will use the relationship we found in Step 4 to find the common tangents: \(y = \pm x \pm \sqrt{a^2 + b^2}\) Comparing this equation with the given options, we see that it matches option (c). Thus, the correct answer is: (c) \(y = \pm x \pm \sqrt{a^2 + b^2}\)

Key concepts

Hyperbola Tangent Equations

Understanding the tangent line equations for a hyperbola is crucial in conic section problems, like those found in JEE advanced mathematics. A tangent to a hyperbola at any point \(x_1, y_1\)\ can be defined using the substitution method where \(x^2\)\ is replaced by \(xx_1\)\ and \(y^2\)\ by \(yy_1\)\ given the hyperbola's standard equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)\ or its conjugate form.
For example, the general equation of a tangent to the first given hyperbola is derived as \(\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1\)\. These tangent equations not only represent straight lines but are crucial in finding common tangents to two hyperbolas, which is often a point of investigation in higher-level mathematics.

Slope of Tangent

The slope is a measure of how steep a line is and is especially significant when dealing with tangents to hyperbolas. For the equation of a straight line, \(y = mx + c\)\, \(m\)\ is the slope. In the context of hyperbolas, the slope of the tangent can be deduced by differentiating the equation of the hyperbola or by manipulating the tangent's equation into the slope-intercept form.
A fascinating aspect comes to light when exploring common tangents to two hyperbolas: if one hyperbola is the conjugate of the other, the slopes of their tangents are negative reciprocals, meaning that the product of the slopes is -1. This property is integral to determining the specific form of the common tangents, as showcased in the example problem.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The hyperbolas studied in the question are just one of the four classic types of conic sections, which also include ellipses, parabolas, and circles. Each conic has a unique set of properties and equations that describe its shape.
The importance of conic sections lies in their wide application across different fields such as astronomy, engineering, and physics. They are foundational concepts in mathematics that provide students with tools to model and solve real-world problems. Hyperbolas, in particular, are known for their asymptotic behavior and are used to describe the trajectories in hyperbolic orbits and in the design of certain optical systems.

JEE Advanced Mathematics

JEE Advanced, one of the most challenging engineering entrance exams in India, features complex problems from various mathematical topics including conic sections. A deep comprehension of hyperbolas and their tangent equations is essential for success in this examination.
Students preparing for JEE Advanced need to develop a thorough understanding of the properties and equations governing conic sections. This includes mastering tasks such as finding the equation of tangents and normals, determining the condition for tangency, and solving problems involving common tangents to multiple conic sections. Problems similar to the one discussed in the textbook exercise exemplify the analytical and problem-solving skills needed for excelling in JEE Advanced mathematics.