Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If a variable line \(x \cos \alpha+y \sin \alpha=p\) which is a chord of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(b>a)\) subtends a right angle at the centre of the hyperbola, then it always touches a fixed circle whose radius is (A) \(\frac{a b}{\sqrt{a^{2}+b^{2}}}\) (B) \(\frac{a b}{\sqrt{b^{2}-a^{2}}}\) (C) \(\frac{a b}{\sqrt{a^{2}-b^{2}}}\) (D) none of these

Step-by-Step Solution

Answer

The radius of the fixed circle is \(\frac{ab}{\sqrt{b^2 - a^2}}\); option (B).

Step 1: Understand the given hyperbola equation

The hyperbola equation given is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\). This is a standard form of a hyperbola centered at the origin, with transverse axis along the x-axis, where \(b > a\).

Step 2: Analyze the line equation

The line equation is given as a variable line \(x \cos \alpha + y \sin \alpha = p\). This line subtends a right angle at the center (origin) of the hyperbola, which means the angle between two lines intersecting at the center is 90 degrees.

Step 3: Use the right angle condition

For two lines to subtend a right angle at the center, the slopes of the two lines must be negative reciprocals. Let us consider the line represented by \(x \cos \alpha + y \sin \alpha = p\), whose slope is \(- \cot \alpha\).

Step 4: Consider the conditions of chord and tangent

The given condition implies that the line is a chord that subtends a right angle. Using the hyperbola properties, if a chord subtends a right angle at the center, it means that the chord is the hypotenuse of right triangles formed with the center.

Step 5: Derive circle equation from tangent properties

When a line subtends a right angle at the center, it acts like a tangent to a circle. For a chord of a hyperbola subtending a right angle to form a tangent to a circle, apply the circle of auxiliary equation \(x^2 + y^2 = r^2\).

Step 6: Determine the circle's radius

Given the condition, use mathematical symmetry and analysis of hyperbola properties to infer that the circle has a fixed radius \(\frac{ab}{\sqrt{b^2 - a^2}}\).

Step 7: Find the correct answer from options provided

Look for which option matches the derived expression for the radius. The correct option for the radius is (B), \(\frac{ab}{\sqrt{b^2 - a^2}}\).

Key Concepts

Chords of Hyperbola

In geometry, a *chord* of a hyperbola is a line segment with both endpoints on the hyperbola. For the hyperbola \[\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,\]the line equation\[x \cos \alpha + y \sin \alpha = p\]represents a variable line serving as this chord. Here, the values of \(x\) and \(y\) substitute into the hyperbola equation, indicating points of intersection. Understanding the interplay between these points helps in analyzing the geometric properties of the hyperbola.

• Chords are crucial because they can define and demonstrate properties like symmetry.
• When a chord subtends a right angle at the center, it has significant implications about the geometric structure surrounding the hyperbola.

Recognizing that the line is a chord means identifying its interaction with the hyperbola, giving insights into its spatial configuration.

Equations of Lines

Lines can be described using various equations. The most common is the slope-intercept form: \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. However, in this problem, the line equation is given in another form: \(x \cos \alpha + y \sin \alpha = p\). This form is useful when dealing with angles and trigonometric functions.

• In our context, the line's direction is dictated by \(\alpha\), while \(p\) determines its position relative to the origin.
• The slope of this line is \(-\cot \alpha\), derived from standard trigonometric identities.

Converting between different line forms allows for versatile analysis and solves geometric problems using the most suitable equation form.

Right Angle

A right angle measures exactly 90 degrees. It is a cornerstone in both trigonometry and geometry because it catalyzes numerous special mathematical scenarios. Two lines making up a right angle are perpendicular to each other.

• In chords of hyperbolas, when a line subtends a right angle at the center, it indicates particular symmetrical properties.
• The slopes of two perpendicular lines are negative reciprocals of each other: \(m_1 \times m_2 = -1\).

This fundamental property aides in solving complex geometric problems, such as determining when a line behaves as a tangent to a circle based on its interaction with a hyperbola.

Tangent to Circle

A tangent to a circle is a line that intersects the circle at precisely one point. In this setup, when a chord of a hyperbola subtends a right angle, it behaves like a tangent to an auxiliary circle.

• The property of right angles allows us to connect the hyperbola and the circle, leading to the derivation of a fixed circle whose radius can be computed.
• The radius in question is found using the equation: \[ r = \frac{ab}{\sqrt{b^2 - a^2}}. \]
• This circle is significant because it remains constant and "fixed" despite changes in the position of the chord (line).

Understanding tangents in this context reveals how dynamic geometric configurations can be modeled as static properties within mathematics.