Problem 24 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The line \(\mathbf{r}=\mathbf{a}+t \mathbf{b}\) touches the sphere \(\mathbf{r}^{2}-2 \mathbf{r} \cdot \mathbf{c}+\mathbf{h}=\) \(0, c^{2}>h\) at the point with position vector \(a\) if (A) \((\mathrm{a}-\mathbf{b}) \cdot \mathbf{c}=0\) (B) \((\mathbf{a}-\mathbf{c}) \cdot \mathbf{b}=0\) (C) \((\mathbf{b}-\mathbf{c}) \cdot \mathbf{a}=0\) (D) \(\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}=0\)

Step-by-Step Solution

Answer

Option (B) \((\mathbf{a} - \mathbf{c}) \cdot \mathbf{b} = 0\) is correct.

Step 1: Understanding the Problem

We are given a line that touches a sphere at a specific point. This means the line is tangent to the sphere at point \( \mathbf{a} \). The equation for the line is \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \), and the equation for the sphere is \( \mathbf{r}^2 - 2 \mathbf{r} \cdot \mathbf{c} + \mathbf{h} = 0 \). The condition \( c^2 > h \) indicates that we have a proper sphere (not imaginary). Our goal is to find which condition among the options ensures the given tangency.

Step 2: Expressing the Sphere's Center and Radius

The center of the sphere is \( \mathbf{c} \) and its radius \( R \) can be derived from the sphere's equation as \( R = \sqrt{c^2 - h} \). The line touches the sphere indicates that the perpendicular distance from the center \( \mathbf{c} \) to the line must be equal to the radius \( R \).

Step 3: Condition for Tangency - Distance Formula

The distance \( d \) from the center \( \mathbf{c} \) to the line \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \) can be calculated using the formula \( d = \frac{|(\mathbf{a} - \mathbf{c}) \cdot \mathbf{b}|}{\| \mathbf{b} \|} \). Since the given line is a tangent, this distance equals the radius \( R \).

Step 4: Analyzing Options

Since the distance formula simplifies to \((\mathbf{a} - \mathbf{c}) \cdot \mathbf{b} = 0\) for tangency (the numerator before taking absolute value), we find that this is precisely option (B). This means our line is perpendicular to the vector \( \mathbf{a} - \mathbf{c} \), ensuring the tangency condition is fulfilled.

Key Concepts

Line Geometry

Understanding the geometry of a line is a fundamental aspect of learning how it interacts with other geometrical objects, like spheres. A line in vector form can be expressed as \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \). Here:

• \( \mathbf{a} \) is a fixed point on the line, known as the position vector.
• \( \mathbf{b} \) is the direction vector of the line. It dictates the line's direction and slope.
• \( t \) is a scalar parameter, which when varied, traces out the entire line.

In the context of our problem, the line is described with these vectors, revealing how it passes through space and at what trajectory. Grasping this helps us understand constraints and conditions for tangency with other forms, like a sphere.

Sphere Equation

Spheres in 3D geometry are defined by their center and radius. The equation \( \mathbf{r}^2 - 2 \mathbf{r} \cdot \mathbf{c} + \mathbf{h} = 0 \) represents a sphere, where:

• \( \mathbf{c} \) is the center of the sphere.
• \( \mathbf{h} \) is a constant linked to the sphere's radius \( R = \sqrt{c^2 - h} \).

The notion \( c^2 > h \) ensures that our sphere is a real sphere (not imaginary), as the square root provides a real, positive value. Understanding how this equation is configured helps interpret where and how lines, planes, or other spheres might meet or intersect with this spherical boundary.

Tangency Condition

When a line is tangent to a sphere at a specific point, it means that the line touches the sphere at exactly one point. The tangency condition is crucial as it ensures the line never actually enters the interior of the sphere.To find the tangency condition, we consider the distance. The formula \( d = \frac{|(\mathbf{a} - \mathbf{c}) \cdot \mathbf{b}|}{\| \mathbf{b} \|} \) is key. It computes the perpendicular distance from the sphere's center \( \mathbf{c} \) to the line \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \).

• For tangency, this distance equals the sphere's radius \( R \).
• If simplified, it leads to the condition \((\mathbf{a} - \mathbf{c}) \cdot \mathbf{b} = 0\).

This suggests that \( \mathbf{b} \) is perpendicular to the vector \( \mathbf{a} - \mathbf{c} \), fulfilling the tangency criterion.