Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

The locus of a point \(P(\alpha, \beta)\) moving under the condition that the line \(y=\alpha x+\beta\) is a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is (A) an ellipse (B) a circle (C) a parabola (D) a hyperbola

Step-by-Step Solution

Answer

The locus is a hyperbola.

Step 1: Equation of Tangent to Hyperbola

The equation of the tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) can be written as \( y = mx \pm \sqrt{a^2m^2-b^2} \). This equation represents a line with slope \(m\) that is a tangent to the hyperbola.

Step 2: Tangent Form Comparison

Given that the line \( y = \alpha x + \beta \) is tangent to the hyperbola, it should match the form \( y = mx + c \) with \( c = \pm \sqrt{a^2m^2-b^2} \). Here, \( \alpha \) corresponds to \( m \) and \( \beta \) to \( c \), meaning \( \beta = \pm \sqrt{a^2 \alpha^2 - b^2} \).

Step 3: Set up the Locus Equation

Rearrange the equation \( \beta = \pm \sqrt{a^2 \alpha^2 - b^2} \) to find the relationship between \( \alpha \) and \( \beta \). Squaring both sides, we get \( \beta^2 = a^2 \alpha^2 - b^2 \). Rearranging gives the locus equation: \( \frac{\alpha^2}{b^2} - \frac{\beta^2}{a^2} = 1 \).

Step 4: Identify the Conic Section

The equation \( \frac{\alpha^2}{b^2} - \frac{\beta^2}{a^2} = 1 \) is in standard form for a hyperbola. Therefore, the locus of point \( P(\alpha, \beta) \) is a hyperbola.

Key Concepts

Locus

In geometry, a **locus** is a collection of points that share a specific property or satisfy a particular condition. In this exercise, the locus represents all points \( P(\alpha, \beta) \) such that the line \( y = \alpha x + \beta \) is tangent to a given hyperbola. A locus is determined through equations that identify all possible positions meeting the criteria imposed by geometrical settings.

• To find the locus, we start by recognizing the conditions that the line must satisfy to be tangent to the hyperbola.
• Specifically, we manipulate these conditions to express the coordinates \( (\alpha, \beta) \) as a locus equation.

In our case, the process involved solving the equation \( \beta = \pm \sqrt{a^2\alpha^2 - b^2} \), leading us to the hyperbolic form \( \frac{\alpha^2}{b^2} - \frac{\beta^2}{a^2} = 1 \). The resulting equation describes a hyperbola, showing how conditions on a moving point can be systematically analyzed through loci.

Tangent to Hyperbola

Understanding how a line interacts with a hyperbola is key in solving this problem. A **tangent to a hyperbola** touches the conic at exactly one point, maintaining an important mathematical property of having equal normal and tangent segments.

• The general form of a tangent line to a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is \( y = mx \pm \sqrt{a^2m^2-b^2} \).
• The condition for tangency is that the line should fulfill \( c = \pm \sqrt{a^2m^2-b^2} \).

In this context, given a line \( y = \alpha x + \beta \) being tangent, the slope \( m \) matches \( \alpha \), and \( \beta \) must equal \( \pm \sqrt{a^2\alpha^2-b^2} \). From this understanding, you derive the necessary criteria for tangency and use them to construct the locus equation.

Conic Sections

**Conic sections** are curves obtained as intersections of a plane with a double-napped cone, classified into four types: circles, ellipses, parabolas, and hyperbolas. Each has distinct equations and properties that define its geometry. Hyperbolas, part of this family, arise when the plane cuts through both nappes of the cone.

• The standard equation for a hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), showing its open curves and asymptotics.
• Hyperbolas differ from ellipses and circles due to their focal distances and the nature in which their points diverge.

The exercise demonstrated identifying a hyperbolic locus through its equation \( \frac{\alpha^2}{b^2} - \frac{\beta^2}{a^2} = 1 \), which precisely follows from the conditions set by the tangency line and confirms the geometric characteristics of hyperbolas among the conic sections. This illustrates the intimate connection and interplay between algebraic expressions and geometric shapes found in conics.