Problem 222 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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

(d) a hyperbola

Step 1: Equation of the tangent line

The condition given is that the line \(y = \alpha x + \beta\) is tangent to the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\). The equation of any tangent to the hyperbola can be written as \(y = mx \pm \sqrt{a^2m^2 - b^2}\) where \(m\) is the slope of the tangent line.

Step 2: Match line equation to tangent formula

We know that the line \(y = \alpha x + \beta\) must match the form of the tangent line equation. This implies that \(m = \alpha\) and \(\beta = \pm \sqrt{a^2\alpha^2 - b^2}\).

Step 3: Simplifying the condition

Square both sides of the equation \(\beta = \pm \sqrt{a^2\alpha^2 - b^2}\) to remove the square root: \(\beta^2 = a^2\alpha^2 - b^2\). Rearranging gives \(\beta^2 = a^2\alpha^2 - b^2\).

Step 4: Rewrite in standard form

Rewrite the equation \(\beta^2 = a^2\alpha^2 - b^2\) as \(a^2\alpha^2 - \beta^2 = b^2\). Divide by \(b^2a^2\) to obtain: \(\frac{\alpha^2}{\frac{b^2}{a^2}} - \frac{\beta^2}{b^2} = 1\).

Step 5: Identify the type of conic section

The standard form obtained \(\frac{\alpha^2}{\frac{b^2}{a^2}} - \frac{\beta^2}{b^2} = 1\) matches the general form of a hyperbola \(\frac{x^2}{p^2} - \frac{y^2}{q^2} = 1\). Therefore, the locus of point \(P(\alpha, \beta)\) is a hyperbola.

Key Concepts

Hyperbola

A hyperbola is a fascinating type of conic section characterized by its open curve formed by intersecting a plane with a double cone. Unlike ellipses and circles, hyperbolas consist of two disjoint, mirror-image branches. These branches are symmetrically placed relative to the center of the hyperbola.

• A hyperbola is defined by its two focus points, and any point on the hyperbola has a constant difference in distance to these focus points.
• The general equation for a hyperbola oriented along the x-axis is: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This tells us the relationship between the x and y coordinates for any point on the curve.
• In a hyperbola's equation, if the x terms are subtracted from the y terms, the hyperbola opens along the y-axis instead.

When talking about hyperbolas in the context of a locus, a tangent line condition can guide where the point of tangency lies on the hyperbola. It provides a rich structure to study geometric properties and solve related problems in algebra.

Tangent Line

A tangent line to a hyperbola or any curve is a line that intersects the curve at exactly one point. This unique intersection point is called the "point of tangency." With hyperbolas, tangent lines play a crucial role in understanding the curve’s geometry.

• A tangent line only touches a hyperbola at one point, implying at that point, the line has the exact same slope as the curve.
• The equation of the tangent line to a hyperbola can also demonstrate some symmetry, as it involves the slope of the tangent touching the curve and the structure of the hyperbola’s equation.
• Formulating and solving tangent line equations is critical in many applications, such as finding specific points or verifying if a given line is tangent to the hyperbola.

Understanding how to derive and use the tangent line equation can illuminate ways to assess intersections and angles between the hyperbola and various lines.

Conic Sections

Conic sections are shapes derived from slicing a cone at different angles and positions. They include circles, ellipses, parabolas, and hyperbolas. Each of these shapes has unique equations and properties that define their geometric form.

• Circular and elliptical sections are formed through slicing a cone parallel to its base and at an angle less steep than the side of the cone, respectively.
• If cut parallel to one of the sides, the result is a parabola, which has only one point of focus.
• Finally, intersecting the cone with a plane arriving at both nappes creates a hyperbola, showcasing its distinct double-branch structure.

In mathematics, these sections form a cornerstone for study, providing a framework to understand curves and motions modeled by these equations across various fields.

Equation of Tangent

The equation of a tangent is key in identifying if a line is tangent to a conic section such as a hyperbola. This equation changes based on the conic it's applied to.

• For a hyperbola defined by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equation of the tangent line becomes \(y = mx \pm \sqrt{a^2m^2 - b^2}\), where \(m\) is the slope.
• This specific form helps in ensuring that the line only touches at one point, maintaining tangency.
• Equations like these are instrumental not just in theoretical problems but also in practical scenarios, like designing tangent paths in engineering structures.

To apply this, convert given conditions into the tangent form, verify the possibility of a solution, and thus confirm tangentiality to the conic.