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

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

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

Step 1: Understand tangency condition

For a line to be tangent to a sphere, it must touch the sphere at exactly one point. If the line equation is \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \), then at the point of tangency \( t = 0 \) and \( \mathbf{r} = \mathbf{a} \). Therefore, \( \mathbf{a} \) lies on the sphere.

Step 2: Substitute \( \mathbf{a} \) in the sphere equation

Substituting \( \mathbf{r} = \mathbf{a} \) into the sphere equation gives \( \mathbf{a}^2 - 2 \mathbf{a} \cdot \mathbf{c} + \mathbf{h} = 0 \). This confirms that \( \mathbf{a} \) is on the sphere.

Step 3: Determine the condition for tangency using \( \mathbf{b} \)

For the line to be tangent, the direction \( \mathbf{b} \) of the line must be perpendicular to the radius at the point of tangency, i.e., \( \mathbf{a} - \mathbf{c} \). This requires \( (\mathbf{a} - \mathbf{c}) \cdot \mathbf{b} = 0 \).

Step 4: Match with given options

The condition \( (\mathbf{a} - \mathbf{c}) \cdot \mathbf{b} = 0 \) matches option \( (B) \). This is the required condition for the given line to be tangent to the sphere at point \( \mathbf{a} \).

Key Concepts

Tangency Condition

Understanding the tangency condition is an essential aspect of vector geometry, particularly when dealing with spheres and lines. A line tangent to a sphere means that it only touches the sphere at a single, unique point. This is a critical concept because any deviation from a single point contact would mean the line is either secant (crosses through the sphere) or does not intersect the sphere at all.
To determine tangency, we start by inserting the line's equation, generally given as \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \), into the sphere's equation. The essence of the tangency condition comes into play when we set \( t = 0 \), which helps transform the line equation into \( \mathbf{r} = \mathbf{a} \).
Here, the position vector \( \mathbf{a} \) needs to satisfy the sphere's equation, \( \mathbf{r}^2 - 2 \mathbf{r} \cdot \mathbf{c} + \mathbf{h} = 0 \). By substituting \( \mathbf{a} \) into the sphere equation, we confirm that \( \mathbf{a} \) lies on the surface of the sphere. Thus, we ensure the prerequisite for tangency, which establishes that the line is tangent to the sphere at point \( \mathbf{a} \).
Not only does this confirm that the line touches the sphere, but it also solidifies our understanding of how vector equations work at the point of contact.

Line Tangent to Sphere

In vector geometry, when we say a line is tangent to a sphere, it implies that the line grazes the sphere such that it only intersects at one point. To mathematically establish tangency, we look for conditions where the line equation and sphere equation align perfectly at that unique point.
For the line \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \), tangency is verified by both ensuring that \( \mathbf{a} \) lies on the sphere and establishing a perpendicular relationship between the line's direction and the sphere's radius at the point of contact.
The process involves substituting \( \mathbf{a} \) into the sphere’s equation \( \mathbf{r}^2 - 2 \mathbf{r} \cdot \mathbf{c} + \mathbf{h} = 0 \). When this holds true, it confirms the tangency condition. When derived correctly, this condition points to specific criteria within the problem, as in our case, the direction \( \mathbf{b} \) results in option \( (B) \). Overall, the mathematical precision of these steps reveals the robust nature of tangency in vector equations.

Direction Vector Perpendicularity

Direction vector perpendicularity is another key element in verifying tangency in vector geometry. When a line is tangent to a sphere, the line's direction vector, given by \( \mathbf{b} \), must be perpendicular to the radius of the sphere at the tangent point.
Mathematically, this perpendicularity is expressed by the dot product. If the dot product of two vectors equals zero, those vectors are perpendicular. For our problem, the condition can be defined as \( (\mathbf{a} - \mathbf{c}) \cdot \mathbf{b} = 0 \).
Here, \( \mathbf{a} - \mathbf{c} \) is the vector from the sphere's center to the point of tangency. This expression guarantees the perpendicular orientation of the direction vector \( \mathbf{b} \) at the tangent point. Aligning this perpendicular relationship solidifies the condition for tangency and verifies our earlier insertion and value testing. So, whenever the dot product evaluates to zero, it confirms that \( \mathbf{b} \) is correctly perpendicular, validating our tangency condition.