Problem 23 - A Complete Resource Book in Mathematics for JEE Main

Question

In a \(\Delta O A B, \mathrm{E}\) is the mid-point of \(O B\) and \(D\) is a poir on \(A B\) such that \(A D: D B=2: 1 .\) If \(O D\) and \(A E\) intes sect at \(P\), then the ratio \(O P: P D\) is (A) \(1: 2\) (B) \(2: 1\) (C) \(3: 2\) (D) \(2: 3\).

Step-by-Step Solution

Answer

The ratio \( OP:PD \) is 1:2. (Option A)

Step 1: Understanding the Problem

We have a triangle \( \Delta OAB \) with points \( E \) as the midpoint of \( OB \) and \( D \) dividing \( AB \) in the ratio \( 2:1 \). We need to find the intersection ratio of \( OD \) and \( AE \) at \( P \), specifically \( OP:PD \).

Step 2: Analyzing the Geometry

Since \( E \) is the midpoint of \( OB \), the coordinates of the midpoint \( E \) can be expressed using the midpoint formula. We also note that \( D \) divides \( AB \) in a \( 2:1 \) ratio, which means any parameter along \( AB \) can be represented using a segment division formula.

Step 3: Using Vector or Coordinate Approach

To solve this, assign coordinates to points \( O (0,0) \), \( A (2a, 2b) \), and \( B (3a, 3b) \). Calculate \( E \)'s coordinates as the midpoint of \( OB \). Use the section formula for \( D \) with \( AD:DB=2:1 \), finding its coordinates as \( \frac{4a}{3}, \frac{4b}{3} \).

Step 4: Setting Up Equations for Intersection

The line \( OD \) can be expressed as a vector equation from \( O \) to \( D \). Similarly, the line \( AE \) is expressed from \( A \) to \( E \). Set these equations equal to find the intersection point \( P \), implying both parametric segments coincide at \( P \).

Step 5: Solving for Intersection Ratio

Solve for the parameters where \( OD \) and \( AE \) intersect. In the vector approach, solve the simultaneous equations derived from line parameters. After equating and solving these equations, find the proportional value of \( t \) which represents the proportion of \( OP:PD \).

Step 6: Determine the Ratio

After solving the parameter equations, you find that \( t = \frac{1}{3} \), meaning \( OP:PD = 1:2 \). As \( OD \) is divided in this particular ratio, the correct answer is found.

Key Concepts

Midpoint Formula

The midpoint formula is a crucial concept in geometry that helps to find the exact middle point between two given points in a plane. It is widely used to solve problems involving midpoints of line segments. The formula is \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two endpoints of the line segment.

• Apply the formula by summing the x-coordinates and dividing by 2 for the x-value of the midpoint.
• Do the same with the y-coordinates for the y-value.

In this exercise, point \( E \) is the midpoint of \( OB \), so we utilize this formula to find the coordinates of \( E \). It's simple: just substitute the coordinates of \( O \) and \( B \). Understanding how to use the midpoint formula makes solving midpoint problems a breeze.

Section Formula

The section formula is another powerful tool used for dividing a line segment into a particular ratio. It's particularly helpful when you want to find a point \( D \) that divides a line segment in any given ratio \( m:n \). This formula is expressed as \[ D = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the endpoints.

• Use \( m \) and \( n \) to weigh the contributions of each endpoint.
• Find where a line segment's point divides it in that specified ratio.

In \( \Delta OAB\), point \( D \) divides \( AB \) in a \( 2:1 \) ratio using this formula, enabling us to calculate its exact position. This allows precise solutions in geometric problems where division isn't uniform.

Vector Approach

Geometry often involves determining intersections and arrangements of lines, which is where the vector approach shines. It converts geometric problems into algebraic ones, making it easier to solve via coordinates. This involves expressing lines as vectors:- Every direction from the origin \( O \) can be seen as a vector.- By using vectors, such as \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \), you can represent line segments or paths.
A vector approach allows us to define lines \( OD \) and \( AE \) more straightforwardly. We solve the equations that represent these lines to find their intersection point. The unique representation of vectors simplifies solving simultaneous equations to find intersections. This makes this approach highly effective in geometry.

Intersection Ratio

When lines intersect, they often divide each other into segments with specific ratios. The intersection ratio focuses on understanding these proportions. This involves determining how a point \( P \) partitions a segment like \OD in such situations. Here’s the process:- Use vector equations of lines to locate where they meet.- Assess how many parts one segment splits into at the intersection.
In the problem, you find the ratio \( OP:PD = 1:2 \). This means that \( P \) divides \OD with \OP being one part and \PD twice that part.By using mathematical logic, coefficients from the parameter equations become the key. This makes defining the intersection ratio simple and precise, revealing the relational size of segments when lines or paths overlap.