Problem 68 - A Complete Resource Book in Mathematics for JEE Main

Question

The range of values of \(p\) such that the angle \(\theta\) between the pair of tangents drawn from the point \((p, 0)\) to the circle \(x^{2}+y^{2}=1\) lies in \(\left(\frac{\pi}{3}, \pi\right)\) is (A) \((-2,-1) \cup(1,2)\) (B) \((-3,-2) \cup(2,3)\) (C) \((0,2)\) (D) none of these

Step-by-Step Solution

Answer

The answer is (B): \((-3,-2) \cup (2,3)\).

Step 1: Understand the Problem

We need to find the range of values for \( p \) such that the angle \( \theta \) between the tangents from point \((p, 0)\) to the circle \( x^2 + y^2 = 1 \) is in the interval \( \left( \frac{\pi}{3}, \pi \right) \). A pair of tangents can be drawn when the power of the point \((p, 0)\) with respect to the circle is positive, which further depends on the distance of the point from the center of the circle.

Step 2: Calculate Power of Point

The power of the point \((p, 0)\) with respect to the circle \(x^2 + y^2 = 1\) is given by \( p^2 - 1 \). This must be positive for the tangents to exist, so we require \( p^2 - 1 > 0 \). Therefore, \( p^2 > 1 \), meaning \( p > 1 \) or \( p < -1 \).

Step 3: Use the Formula for Angle Between Tangents

The formula for the angle \( \theta \) between two tangents from a point \((p, 0)\) to a circle \(x^2 + y^2 = r^2\) is \( \cos \theta = \frac{r}{\sqrt{x_1^2 + y_1^2}} \). For our circle, \( r = 1 \), so \( \cos \theta = \frac{1}{|p|} \).

Step 4: Determine Range of \(\theta\)

We want \( \frac{\pi}{3} < \theta < \pi \), meaning \( \frac{1}{2} > \cos \theta > -1 \). Therefore, \( |p| > 2 \) when \( \cos \theta < \frac{1}{2} \). Hence, \( p > 2 \) or \( p < -2 \).

Step 5: Identify Valid Range for \(p\)

From Steps 2 and 4, the valid range for \( p \) is \( p > 2 \) or \( p < -2 \), which fulfills both the point power condition and the angle condition. This results in the solution \((-\infty, -2) \cup (2, \infty)\).

Step 6: Compare with Given Options

The range from the options closest to our solution \((-\infty, -2) \cup (2, \infty)\) is (B) \((-3,-2) \cup (2,3)\).

Key Concepts

Power of a point

When analyzing the angle between tangents drawn from a point to a circle, the "power of a point" is a crucial concept. It essentially measures how a given point relates algebraically to a circle. This power is calculated from the equation of a circle and provides insight into whether tangents from the point to the circle can actually exist.
For a circle centered at the origin, like our problem where the equation is given by \( x^2 + y^2 = 1 \), the power of the point \((p, 0)\) is derived from substituting the point into the circle's equation. It results in the equation \( p^2 - 1 \).

• If \( p^2 - 1 > 0 \), the point lies outside the circle, and two tangents can indeed be drawn.
• If \( p^2 - 1 = 0 \), the point is on the circle, and exactly one tangent can be drawn.
• If \( p^2 - 1 < 0 \), the point is inside the circle, and no tangents can be drawn.

Thus, for our tangents to exist, \( p^2 - 1 \) must be positive, suggesting \( p > 1 \) or \( p < -1 \). This sets the groundwork for further exploration into the conditions for angle \( \theta \).

Cosine of angles

The cosine of angles formula is instrumental when calculating the angle between two tangents from a point to a circle. Understanding this formula helps in determining whether the given angular range condition is met.
For the two tangents from the point \((p, 0)\) to the circle \(x^2 + y^2 = 1\), the angle \( \theta \) between these tangents is calculated using the formula: \[\cos \theta = \frac{r}{\sqrt{x_1^2 + y_1^2}} \]where \( r \) is the radius of the circle and \((x_1, y_1)\) is the point from which tangents are drawn. In this case, since \( r = 1 \) and \( (x_1, y_1) = (p, 0) \), it simplifies to:\[\cos \theta = \frac{1}{|p|} \]
Hence, the values of \( \cos \theta \) fall inversely with \(|p|\). For \( \theta \) to be within the range of \( \frac{\pi}{3} < \theta < \pi \), \( \cos \theta \) must be between -1 and \( \frac{1}{2} \). This leads us to the condition that \(|p| > 2\), which we'll delve into further in the context of determining the valid range of \( p \).

Range of values

The range of values in the problem helps us identify which values of \( p \) result in an angle \( \theta \) within the desired range of \( \frac{\pi}{3} < \theta < \pi \). For this, we need to examine these two main conditions:

• First, from the power of a point, \( p^2 - 1 > 0 \) gives \( p > 1 \) or \( p < -1 \). This ensures tangents can exist.
• Then, from the cosine of angles condition, \( \frac{1}{2} > \cos \theta > -1 \) further restricts \( |p| > 2 \).

Combining these two conditions, we find that \( p \) must be either greater than 2 or less than -2 to satisfy both the existence of tangents and the specific angle range. This results in the range \( (-\infty, -2) \cup (2, \infty) \). Given the closest option, the answer is (B): \((-3,-2) \cup (2,3)\).
This identification of values ensures the conditions of both mathematical concepts fit perfectly within the problem.