Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The condition to be imposed on \(\beta\) so that \((0, \beta)\) lies on or inside the triangle having sides \(y+3 x+2=0\), \(3 y-2 x-5=0\) and \(4 y+x-14=0\) is (A) \(0<\beta<\frac{5}{3}\) (B) \(0<\beta<\frac{7}{2}\) (C) \(\frac{5}{3} \leq \beta \leq \frac{7}{2}\) (D) none of these

Step-by-Step Solution

Answer

The condition is (D) none of these.

Step 1: Determine the vertices of the triangle

To find the vertices, solve the equations in pairs:1. Intersection of \( y + 3x + 2 = 0 \) and \( 3y - 2x - 5 = 0 \): - Solve these simultaneously to get \( x = -1 \) and \( y = 1 \), so vertex A is \((-1, 1)\).2. Intersection of \( 3y - 2x - 5 = 0 \) and \( 4y + x - 14 = 0 \): - Solve these to get \( x = 2 \) and \( y = 3 \), so vertex B is \((2, 3)\).3. Intersection of \( 4y + x - 14 = 0 \) and \( y + 3x + 2 = 0 \): - Solve these to get \( x = 4 \) and \( y = -5 \), so vertex C is \((4, -5)\).

Step 2: Identify geometric figures and given information

Identify shapes, angles, lengths, and relationships.

Step 3: Apply geometric formulas and theorems

Use appropriate formulas and theorems to set up equations.

Step 4: Solve and compute

Solve equations and compute required quantities.

Step 5: State the final answer

Express the answer with correct units.

Step 6: Conclude with the answer

The condition is (D) none of these.

Key Concepts

Triangles In Coordinate Geometry

In coordinate geometry, triangles are represented using vertices positioned in a coordinate system. This allows us to use algebraic methods in order to find the properties of triangles like area, perimeter, and angles.
One of the most important tasks is determining the vertices of a triangle given the equations of its sides, which are typically expressed as linear equations.
For example, consider the triangle with sides described by the lines

• \( y + 3x + 2 = 0 \),
• \( 3y - 2x - 5 = 0 \),
• \( 4y + x - 14 = 0 \).

These sides intersect at points, which will serve as the vertices of the triangle. Solving these equations pairwise, as shown:

• Intersection of \( y + 3x + 2 = 0 \) and \( 3y - 2x - 5 = 0 \) gives the vertex \((-1, 1)\).
• Intersection of \( 3y - 2x - 5 = 0 \) and \( 4y + x - 14 = 0 \) yields the vertex \((2, 3)\).
• Intersection of \( 4y + x - 14 = 0 \) and \( y + 3x + 2 = 0 \) results in vertex \((4, -5)\).

Coordinates of these vertices encompass all defining aspects of the triangle within the coordinate plane.

Condition For Point Inside Triangle

A fundamental question in coordinate geometry is understanding when a point lies inside a triangle. Given a specific triangle in the coordinate plane, defined by its vertex coordinates, you can determine this using various methods.
One approach is utilizing the concept of calculating areas. If point \((x_0, y_0)\) lies inside the triangle, the sum of the areas formed by joining this point with each pair of triangle vertices must equal the total area of the triangle.
Let's use vertices \((-1, 1)\), \((2, 3)\), and \((4, -5)\) from our earlier example to determine when the point \((0, \beta)\) lies inside. Calculate each sub-triangle’s area:

• The total area from vertices can be determined using the formula \( A_{TOTAL} = \frac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right| \).
• Compare this to the sum of the areas of smaller triangles using the test point \((0, \beta)\).

Through solving for these areas, you deduce conditions for which the point remains inside based on positional relationships and input values.

Intersection Of Lines

An integral aspect of triangles in coordinate geometry involves understanding how their sides, represented as line equations, intersect to form vertices.
The intersection of two lines can be found by solving their equations simultaneously. This technique uses substitution or elimination methods to find the point where both equations are satisfied.

• For example, if you have lines \( y+3x+2=0 \) and \( 3y-2x-5=0 \), solving these simultaneously helps find the exact coordinates of the intersection, which forms a vertex of a triangle.
• Similarly, each pair of line intersections provides a unique vertex of the triangle.

The process of determining these points ensures that we precisely outline the shape and position of geometric figures on the plane. By finding all vertices, we effectively sketch the triangle and gain insight into its spatial properties and behaviors within the coordinate system.