Problem 43 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of point(s) outside the hyperbola \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\) from where two perpendicular tangents can be drawn to the hyperbola is/are (A) none (B) 1 (C) 2 (D) infinte

Step-by-Step Solution

Answer

(D) infinite

Step 1: Understand hyperbola equation

The given hyperbola is \(\frac{x^2}{25} - \frac{y^2}{36} = 1\), which is centered at the origin \((0,0)\), with its transverse axis along the x-axis.

Step 2: Condition for perpendicular tangents

For a point outside a hyperbola, if perpendicular tangents can be drawn, then the point must lie on the director circle of the hyperbola. This circle is described by the equation \(x^2 + y^2 = a^2 + b^2\).

Step 3: Calculate the director circle

For the given hyperbola, \(a^2 = 25\) and \(b^2 = 36\). Therefore, the director circle is given by \(x^2 + y^2 = 25 + 36 = 61\).

Step 4: Nature of the director circle

The director circle is a locus of points where the perpendicular tangents to the hyperbola can be drawn. Any point that lies on this circle is valid for drawing perpendicular tangents.

Step 5: Conclusion on number of points

The director circle provides an infinite number of points from where perpendicular tangents can be drawn, as each point on the circle meets the condition given.

Key Concepts

Perpendicular Tangents

Perpendicular tangents are lines that can be drawn from a point outside a conic section, touching the curve at an exact right angle to each other. In the context of a hyperbola, such as the one given in the problem, these tangents play a vital role.
For any external point to have perpendicular tangents to a hyperbola, it must satisfy a specific condition: the point should lie on the hyperbola's director circle. This ensures that both tangents intersect the hyperbola at an angle of 90 degrees.

• The point must be external to the hyperbola.
• The tangents intersect each other at 90 degrees.

Understanding this relationship helps expand the knowledge of hyperbolas and how various points outside interact with their geometric structure.

Director Circle

The director circle of a hyperbola is an essential geometric concept related to the drawing of perpendicular tangents. It is the locus of all points from where two perpendicular tangents can be drawn to the hyperbola.
For the specific hyperbola \(\frac{x^2}{25} - \frac{y^2}{36} = 1\), the equation of the director circle is found by calculating the sum of the squares of the hyperbola’s semi-major and semi-minor axes. This results in the circle's equation: \(x^2 + y^2 = 61\).

• Director circle is always larger than the hyperbola.
• It provides infinite points for drawing perpendicular tangents.

Such a circle envelops the hyperbola and signifies the boundary where the special perpendicular tangent condition holds true for all its points.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane in various ways, resulting in shapes like circles, ellipses, parabolas, and hyperbolas. Each conic section has distinct characteristics and properties.
The hyperbola in the problem represents one of these conic sections. It has two symmetrical branches that open in opposite directions along the x-axis, defined by the equation given. Important properties include:

• It is defined by the difference of distances from any point on it to the two foci being constant.
• It showcases asymptotes which are lines the curve approaches but never touches.

By exploring or working with these sections, we gain insight into varied mathematical principles and the natural symmetry of geometric figures.