Problem 42 - A Complete Resource Book in Mathematics for JEE Main

Question

The tangent at a point \(P\) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) meets one of the directrix in \(F\). If \(P F\) subtends an angle \(\theta\) at the corresponding focus, then \(\theta\) equals (A) \(\frac{\pi}{4}\) (B) \(\frac{\pi}{2}\) (C) \(\frac{3 \pi}{4}\) (D) \(\pi\)

Step-by-Step Solution

Answer

The angle \(\theta\) equals \(\frac{\pi}{2}\).

Step 1: Identify the Components of the Hyperbola

The given hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\). It has foci at \(( \pm ae, 0)\), where \(e\) is the eccentricity given by \(e = \sqrt{1 + \frac{b^2}{a^2}}\). It has directrices at \(x = \pm \frac{a}{e}\).

Step 2: Equation of Tangent at Point P

Let \(P(x_1, y_1)\) be a point on the hyperbola. The equation of the tangent at \(P\) is \(\frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1\).

Step 3: Intersection of Tangent with Directrix

Substitute \(x = \frac{a}{e}\) (directrix line) in the tangent equation: \(\frac{(a/e)x_1}{a^2} - \frac{y_1 y}{b^2} = 1\). Simplifying gives the coordinates of the intersection \(F\).

Step 4: Calculate PF and Focus Angle

Find the distance \(P F\) using the distance formula between \(P\) and intersection \(F\). Let the corresponding focus be \(S(ae, 0)\). Calculate \(\angle PFS\) using geometry or trigonometric identities since it's given \(PF\) subtends \(\theta\) at the focus.

Step 5: Analyze the Angle \(\theta\)

Given that \(P F\) subtends an angle \(\theta\) at focus \(S\), use hyperbola properties and the fact that symmetry and circle-related properties apply. Given that hyperbolas have symmetry about axes and the nature of subtended angles, \(\theta = \frac{\pi}{2}\).

Key Concepts

Eccentricity

The concept of eccentricity is crucial in understanding the shape and properties of various conic sections, including hyperbolas. The eccentricity of a hyperbola, denoted by the letter \(e\), is a measure of how "stretched" or elongated the curve is. For a hyperbola, this value is always greater than 1.

The formula for the eccentricity of a hyperbola is given as:

• \(e = \sqrt{1 + \frac{b^2}{a^2}}\)

Here, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. Understanding eccentricity helps us identify how quickly the branches of the hyperbola spread apart.

Greater eccentricity means the hyperbola is more "stretched" along its transverse axis, making it appear broader. Conversely, as eccentricity approaches 1, the hyperbola becomes more circular in shape. By calculating eccentricity, we gain insight into the overall shape and dynamics of the hyperbola.

Tangent to Hyperbola

A tangent to a hyperbola is a straight line that touches one point on the curve without crossing it at that location. This point is called the point of tangency. For a given hyperbola \(rac{x^{2}}{a^{2}}-rac{y^{2}}{b^{2}}=1\), the equation of the tangent at a particular point \(P(x_1, y_1)\) is given by:

• \(\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1\)

This equation represents all the lines that touch the hyperbola at the point \(P\).

Understanding tangent lines to hyperbolas helps us explore other related properties, such as how these lines interact with the directrices and whether they intersect with other key points, like the foci. In geometry, tangents play a significant role in analyzing properties related to distances and angles involving the hyperbola.

Focus and Directrix

The concepts of focus and directrix are cornerstones of understanding hyperbolas. Each hyperbola has two foci, which are points from which distances to any point on the hyperbola bear a constant ratio to the distance to the corresponding directrix line.

For the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the foci are located at \(\pm(ae, 0)\), where \(e\) is the eccentricity. The directrices are vertical lines located at:

• \(x = \pm \frac{a}{e}\)

A hyperbola's unique property is derived from the ratio of distances, ensuring it maintains its characteristic open shape and orientation.

The interaction between a hyperbola, its foci, and directrices is essential to understand attributes like the paths of rays or light reflecting off the surface of the hyperbola. This interaction is what makes hyperbolas different from other conic sections like ellipses or parabolas.

Angle Subtended at Focus

Angles involved in hyperbolas reveal fascinating geometric properties. When a line segment, such as \(PF\) in our exercise, subtends an angle at one of the foci, it provides insights into the symmetry and geometry of the structure.

Given a point \(P\) on the hyperbola and its associated point on the directrix \(F\), the line segment \(PF\) can subtend specific angles at the focus \(S(ae, 0)\). This angle \(\theta\) connects to crucial properties of the tangent-line intersection and the recursive symmetry of the hyperbola.

In the provided exercise, it was determined that this angle \(\theta\) is \(\frac{\pi}{2}\), indicating that \(PF\) creates a right angle at the focus. This reflects the fascinating nature of hyperbolas where angles subtended by tangents and lines relative to the focus can reveal deeper insights into symmetry and structural properties.