Problem 18 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(P(x, y, z)\) is a point on the line segment joining \(Q(2,\), \(2,4)\) and \(R(3,5,6)\) such that the projection of \(O P\) on the axes are \(\frac{13}{5}, \frac{19}{5}, \frac{26}{5}\) respectively, then \(P\) divides \(Q R\) in the ratio (A) \(1: 2\) (B) \(3: 2\) (C) \(2: 3\) (D) \(1: 3\)

Step-by-Step Solution

Answer

The point \(P\) divides the line segment \(QR\) in the ratio \(3:2\).

Step 1: Understanding the Problem

A point \(P(x, y, z)\) divides the line segment joining \(Q(2, 2, 4)\) and \(R(3, 5, 6)\). The projection of \( OP \) on the axes are \( \frac{13}{5}, \frac{19}{5}, \frac{26}{5} \). We need to find the ratio in which \(P\) divides \(QR\).

Step 2: Write Equations for the Projections

Since the projections of \(OP\) are given, \(x = \frac{13}{5}\), \(y = \frac{19}{5}\), and \(z = \frac{26}{5}\). Use these to express \(P(x,y,z)\) and note their coordinates.

Step 3: Determining the Ratio

The coordinates \((x,y,z)\) of \(P\) are found using the section formula. If \(P\) divides \(QR\) in the ratio \(m:n\), then:\[x = \frac{m \, \cdot \, 3 + n \, \cdot \, 2}{m + n}\]\[y = \frac{m \, \cdot \, 5 + n \, \cdot \, 2}{m + n}\]\[z = \frac{m \, \cdot \, 6 + n \, \cdot \, 4}{m + n}\]Substitute the known coordinates of \(P\) into these equations.

Step 4: Setting Up the Equations

Using \( x = \frac{13}{5} \), we set: \[\frac{3m + 2n}{m+n} = \frac{13}{5}\]Simplify to find a relationship between \(m\) and \(n\). Similarly, set equations for \(y\) and \(z\) and simplify.

Step 5: Solving the Equations

After substituting, we obtain:1. \( 5(3m + 2n) = 13(m+n) \).2. \( 5(5m + 2n) = 19(m+n) \).3. \( 5(6m + 4n) = 26(m+n) \).Solve these equations to find the values for \(m\) and \(n\).

Step 6: Finding the Ratio

Simplifying the equations, we find from any one of the above (since they are consistent for one value) \(m:n = 3:2\). This implies \(P\) divides \(QR\) in the ratio of 3:2.

Key Concepts

Section Formula

The section formula is a fundamental tool in coordinate geometry, especially when working with line segments. It helps us determine the coordinates of a point that divides a line segment between two other points in a given ratio.

For example, if there's a point \( P(x,y,z) \) dividing a line segment joining points \( A(x_1,y_1,z_1) \) and \( B(x_2,y_2,z_2) \) in a specific ratio \( m:n \), then the coordinates \((x,y,z)\) of \( P \) are given by:\[x = \frac{mx_2 + nx_1}{m+n}\]\[y = \frac{my_2 + ny_1}{m+n}\]\[z = \frac{mz_2 + nz_1}{m+n}\]The formula accommodates both internal and external division, making it versatile and applicable in various scenarios involving collinearity of points.

• Internal Division: When \( P \) lies between \( A \) and \( B \).
• External Division: When \( P \) divides the line extension beyond one endpoint.

Understanding this formula is crucial when analyzing problems or applications in physics and engineering involving midpoint and division points.

Line Segment Division

Dividing a line segment involves understanding how a point creates parts that relate in a specific ratio. For a line segment joining two points like \( Q(2,2,4) \) and \( R(3,5,6) \), the task can be seen as determining the relative percentage of the entire line length attributed to each sub-segment formed by another point \( P \).

The ratio \( m:n \) tells us how the line segment is split: * The number \( m \) represents how much of \( P \) to \( Q \), and * \( n \) quantifies \( P \) to \( R \). When working with ratios in coordinate geometry, it's often about proportions rather than absolute distances.
These concepts are valuable for:

• Geometrical constructions
• Solving problems related to similar triangles
• Scale modeling scenarios

Understanding the line segment division is pivotal in fields requiring precise measurements and relationships between points and lines.

Projection on Axes

Projection in coordinate geometry refers to the shadow or footprint a line or point creates when "dropped" onto another plane, often one of the coordinate axes. For any point \( P(x, y, z) \) in three-dimensional space, its projections on the axes are the respective views when considering the point down from or onto that axis.

The given projections are:
* \( x = \frac{13}{5} \) * \( y = \frac{19}{5} \) * \( z = \frac{26}{5} \)These values signify the coordinates of \( P \) concerning its alignment with the axis planes, providing insightful details into its spatial orientation. The ability to decouple the three-dimensional positioning into 2D perspectives has applications in:

• Vector analysis
• Physics
• Projection systems such as CAD

Mastering projections helps in simplifying complex three-dimensional problems into manageable one-dimensional components.

3D Coordinates

Three-dimensional coordinates extend the concept of a regular 2D coordinate plane by adding depth to the description of a point's location.

When describing a point \( P(x, y, z) \) in space, each of the values represents:

• "\(x\)" - the horizontal distance along the x-axis
• "\(y\)" - the vertical distance along the y-axis
• "\(z\)" - the depth distance along the z-axis

This extra dimension offers more versatility and realism in graphical representation, necessary for understanding geology, astronomy, and engineering designs.
The three values work in harmony, enabling us to model objects, locate points, or describe shapes in a defined spatial expanse. With 3D coordinates, you can begin exploring rotational movements, trajectory plotting, and even rendering visual graphics on computers, bridging mathematics with real-world simulations.