Problem 40 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{A}(-3,2)\) and \(\mathrm{B}(-2,1)\) be the vertices of a triangle \(\mathrm{ABC}\). If the centroid of this triangle lies on the line \(3 x+4 y+2=0\), then the vertex \(C\) lies on the line : [Online April 25, 2013] (a) \(4 x+3 y+5=0\) (b) \(3 x+4 y+3=0\) (c) \(4 x+3 y+3=0\) (d) \(3 x+4 y+5=0\)

Step-by-Step Solution

Answer

The vertex C lies on the line \(3x + 4y + 3 = 0\), which is option (b).

Step 1: Understand the Problem

We have points A and B of the triangle with coordinates (-3, 2) and (-2, 1) respectively, and we know the centroid of the triangle ABC lies on the line \(3x + 4y + 2 = 0\). We need to find an equation that describes the possible location of vertex C.

Step 2: Apply the Centroid Formula

The formula for the centroid (G) of a triangle with vertices \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\) is given by \(G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\). Substituting the known coordinates for A and B, the centroid coordinates become \(G\left(\frac{-3 - 2 + x_3}{3}, \frac{2 + 1 + y_3}{3}\right)\).

Step 3: Substitute the Centroid Information into the Line Equation

Since the centroid \(G\) lies on the line \(3x + 4y + 2 = 0\), substitute the x and y values of G: \[3\left(\frac{-5 + x_3}{3}\right) + 4\left(\frac{3 + y_3}{3}\right) + 2 = 0\]. Simplify this to get: \(-5 + x_3 + 4 + \frac{4y_3}{3} + 2 = 0\).

Step 4: Simplify the Centroid Equation

Combine like terms in the equation: \(-1 + x_3 + \frac{4y_3}{3} = -2\). Simplify to find \(x_3 + \frac{4y_3}{3} = -1\). Multiply through by 3 to clear the denominator: \(3x_3 + 4y_3 = -3\).

Step 5: Rearrange the Equation to Match Options

Rearrange the equation \(3x_3 + 4y_3 = -3\) to check against the options: divide through by 3 to get \(x + \frac{4}{3}y = -1\) which matches rearranged format of the choices; however, this approach was mistaken. Rewriting the expression directly from \(3x_3 + 4y_3 = -3\) leads to the form \(3x + 4y + 3 = 0\), finding it aligns with option (b).

Step 6: Verify the Solution

Check that if \(3x + 4y + 3 = 0\) corresponds correctly by substituting back into the centroid condition: indeed, it satisfies all respects aligning it as option \((b)\).

Key Concepts

Centroid of a Triangle

The centroid is a key concept in understanding triangle geometry. It is the point where all three medians of the triangle intersect. A median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side.

The centroid has a special property: it is always located at an average point determined by the vertices of the triangle. Mathematically, for a triangle with vertices at \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the coordinate of the centroid \(G\) is given by:\[G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\]This formula calculates the centroid as the average of the x-coordinates and y-coordinates of the triangle's vertices. Using this understanding, you can easily find the balancing point of any triangle, helping in solving various geometric problems.

Equation of a Line

The equation of a line is fundamental in coordinate geometry. It describes all the points that make up a line in a plane. The most common form is the linear equation format: \(ax + by + c = 0\). This describes a line where \(x\) and \(y\) are variables, and \(a, b, \) and \(c\) are constants.

The role of each constant is crucial:

• \(a\) and \(b\) determine the slope of the line, crucial for the line's tilt.
• \(c\) affects where the line sits in the coordinate plane, shifting it up or down.

Understanding this structure helps you identify or construct the equations of lines from points or other geometric conditions like centroids or intersections.

Triangle Vertices

Vertices are the defining points of a triangle in any plane. In coordinate geometry, a triangle is defined by three vertices each with an \(x\) and \(y\) coordinate. For example, a triangle \(ABC\) might have vertices at \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\).

When working with triangles:

• The distance between any two vertices gives you the length of the triangle's sides, computed using the distance formula.
• Knowing the coordinates of the vertices enables calculations of area, perimeter, and various centers of the triangle.

For the problem at hand, understanding the position of each vertex in relation to the centroid and the line equation helps you solve for unknowns, such as the position of a third vertex.