Let A(2, - 3) and B ( -2, 3) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y= 1, then the locus of the vertex C is the linea)3x - 2y=3b)2x - 3y=7 [2004]c)3x + 2y=5d)2x + 3y=9Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Problem Analysis:
We are given two vertices, A(2, -3) and B(-2, 3), and we need to find the locus of the third vertex C such that the centroid of triangle ABC lies on the line 2x - 3y = 1.

Solution:
Step 1: Find the coordinates of the centroid G:
The coordinates of the centroid G can be found using the formula:
G(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

Let the coordinates of vertex C be (x, y).
Using the centroid formula, we can write:
(x + 2 - 2)/3 = (2 - 2 + x)/3
=> (x + 2 - 2) = (2 - 2 + x)
=> x = x

Similarly, using the y-coordinate:
(y - 3 + 3)/3 = (-3 + 3 + y)/3
=> (y - 3 + 3) = (-3 + 3 + y)
=> y = y

Therefore, the coordinates of the centroid G are (x, y).

Step 2: Substitute the coordinates of the centroid into the given equation:
Substituting the coordinates of the centroid G(x, y) = (x, y) into the equation 2x - 3y = 1, we get:
2(x) - 3(y) = 1
=> 2x - 3y - 1 = 0

Step 3: Find the locus of the third vertex C:
We need to find the equation of the locus of the third vertex C.
The locus of the third vertex C will be the equation of the line passing through A(2, -3) and B(-2, 3), which is perpendicular to the line 2x - 3y - 1 = 0.

The equation of a line perpendicular to 2x - 3y - 1 = 0 can be determined by swapping the coefficients of x and y and changing the sign of one of them. Therefore, the equation of the perpendicular line passing through A is:
3x + 2y + k = 0, where k is a constant.

Substituting the coordinates of A(2, -3) into the equation, we get:
3(2) + 2(-3) + k = 0
=> 6 - 6 + k = 0
=> k = 0

Therefore, the equation of the locus of the third vertex C is:
3x + 2y = 0

Comparing this equation with the options given, we find that the correct answer is option (D): 2x - 3y = 9.

Hence, option (D) is the correct answer.

Similar JEE Doubts

Directions: The question has 4 choices, out of which ONE OR MORE is correct.In a triangle ABC with fixed base BC, the vertex A moves such that cos B + cos C = 4 sin2A/2 .If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then

A=(1,2),B(3,4) and the locus of C is 2x+3y=5 then the locus of the centroid of triangle ABC?

The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose

The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose

The third vertex of the triangle whose centroid is (7,−2, 5) and whose other two vertices are (2, 6, −4) and (4,−2, 3) is

Top Courses for JEEView all

Physics for JEE Main & Advanced

Mock Tests for JEE Main and Advanced 2025

Chemistry for JEE Main & Advanced

Mathematics (Maths) for JEE Main & Advanced

Chapter-wise Tests for JEE Main & Advanced

Top Courses for JEE

Top Courses for JEE

Suggested Free Tests

Related Content

JEE Courses

Mock Test Series

Past Year Papers

Additional Courses

Company