Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(P(a \sec \theta, b \tan \theta)\) and \(Q\left(\begin{array}{llll}a & \sec \phi, & b & \tan \phi)\end{array}\right.\) where \(\theta+\phi=\frac{\pi}{2}\), be two points on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 .\) If \((h, k)\) is the point of intersection of the normals at \(P\) and \(Q\), then \(k\) is equal to (A) \(\frac{a^{2}+b^{2}}{a}\) (B) \(-\left(\frac{a^{2}+b^{2}}{a}\right)\) (C) \(\frac{a^{2}+b^{2}}{b}\) (D) \(-\left(\frac{a^{2}+b^{2}}{b}\right)\)

Step-by-Step Solution

Answer

The value of \(k\) is \(-\left(\frac{a^2+b^2}{b}\right)\). Choose (D).

Step 1: Understand the hyperbola equation

The hyperbola is given as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Any point \((x, y)\) on the hyperbola must satisfy this equation.

Step 2: Substitute points on the hyperbola

We have two points, \(P(a \sec \theta, b \tan \theta)\) and \(Q(a \sec \phi, b \tan \phi)\), which must satisfy the hyperbola equation. Substitute these coordinates into the hyperbola equation.

Step 3: Use the identity for complementary angles

Given \(\theta + \phi = \frac{\pi}{2}\), use the trigonometric identity for complementary angles: \(\sec(\frac{\pi}{2} - \theta) = \csc(\theta)\) and \(\tan(\frac{\pi}{2} - \theta) = \cot(\theta)\). This implies \(\sec \phi = \csc \theta\) and \(\tan \phi = \cot \theta\).

Step 4: Calculate normals at P and Q

The equation of the normal at any point \((x_1, y_1)\) on the hyperbola is given by \(\frac{x_1 x}{a^2} - \frac{y_1 y}{b^2} = \frac{x_1^2 - y_1^2}{a^2} + 1\). Use this to find the normals at \(P\) and \(Q\).

Step 5: Set equations for normals

For \(P(a \sec \theta, b \tan \theta)\), the normal equation becomes \(\frac{a \sec \theta \cdot x}{a^2} - \frac{b \tan \theta \cdot y}{b^2} = \left( 1 + \tan^2 \theta \right)\). Similarly, write the normal equation for \(Q\).

Step 6: Find the intersection of the normals

Find the point \((h, k)\) where the normals at \(P\) and \(Q\) intersect. Set the two normal equations equal to each other and solve for \(h\) and \(k\).

Step 7: Solve for k

After solving the equations, you will find \(k = -\left(\frac{a^2+b^2}{b}\right)\).

Step 8: Verify answer choices

Check the calculated \(k\) value against the provided answer choices. The correct choice is (D) \(-\left(\frac{a^2+b^2}{b} \right)\).

Key Concepts

Normals

To understand what a normal is, picture a path or curve. A normal at any point on this curve is a line that is perpendicular to the tangent of the curve at that point. In other words, it's like standing upright at that specific spot on the curve. In mathematical terms, for the hyperbola \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]at a point \((x_1, y_1)\), the equation of the normal is given by:\[\frac{x_1 x}{a^2} - \frac{y_1 y}{b^2} = \frac{x_1^2 - y_1^2}{a^2} + 1\]Turning this equation into a useful tool involves substituting the coordinates of our points, such as \((a \sec \theta, b \tan \theta)\) for point \(P\). By applying the formula, we can find the normals at the points of interest, helping us track where these lines intersect. This systemization is crucial when dealing with geometric configurations on a hyperbola.

Complementary Angles

Complementary angles are two angles whose sum is \(\frac{\pi}{2}\) radians, or 90 degrees. In the context of this exercise, the angles \( \theta \) and \( \phi \) are complementary because their sum is \( \theta + \phi = \frac{\pi}{2} \).Complementary angles hold interesting trigonometric relationships that simplify various calculations. These include:

• \(\sec(\frac{\pi}{2} - \theta) = \csc(\theta)\)
• \(\tan(\frac{\pi}{2} - \theta) = \cot(\theta)\)

Using these identities, we can easily express \( \sec \phi \) as \( \csc \theta \) and \( \tan \phi \) as \( \cot \theta \). This transformation helps in transforming the initial coordinates into a form that is easier to analyze on the hyperbola, and it is essential for calculating normals at these specific points.

Trigonometric Identities

Trigonometric identities are formulas that relate different trigonometric functions to one another. They are like shortcuts that make solving complex equations easier. In the context of the given problem, where we analyze hyperbola and points such as \(P(a \sec \theta, b \tan \theta)\) and \(Q(a \sec \phi, b \tan \phi)\), these identities are invaluable.Some commonly used trigonometric identities include:

• \(\tan^2 \theta + 1 = \sec^2 \theta\)
• \(\sec(\frac{\pi}{2} - \theta) = \csc(\theta)\)
• \(\tan(\frac{\pi}{2} - \theta) = \cot(\theta)\)

These identities enable us to express functions in terms of other angles, particularly complementary ones as discussed previously. By employing these identities, you can simplify the equations involved, make substitutions, and drive towards finding solutions efficiently. For instance, in the problem, they are crucial for transforming and simplifying the points' coordinates to effectively solve for the intersection of the normals.