It is desred to construct a right angled triangle ABC(angle C=π/2) in xy-plane so that its sides are parallel to its coordinate axes and the medians through A and B lie on the lines y=3x+1 and y=mx+2 respectively.The values of 'm' for which such a triangle is possible is/are

a.-12

b.3/4

c.4/3

d.1/12

To solve the problem of constructing a right-angled triangle ABC in the xy-plane with specific conditions on the medians, we need to analyze the geometry involved. The triangle has a right angle at point C, and its sides are aligned with the coordinate axes. This means that if we place point C at the origin (0, 0), point A will lie on the x-axis, and point B will lie on the y-axis. Let's denote the coordinates of the points as follows:

• A = (a, 0)
• B = (0, b)
• C = (0, 0)

Next, we need to find the midpoints of the sides of the triangle to determine the medians. The midpoint M_A of side BC (from B to C) is given by:

M_A = ((0 + 0)/2, (b + 0)/2) = (0, b/2)

Similarly, the midpoint M_B of side AC (from A to C) is:

M_B = ((a + 0)/2, (0 + 0)/2) = (a/2, 0)

The median from A to M_B can be expressed as the line connecting point A to midpoint M_B. The slope of this median is:

Slope of median from A to M_B = (0 - 0) / (a/2 - a) = 0

This means that the median from A is horizontal, which does not match the line y = 3x + 1. Therefore, we need to focus on the median from B to M_A.

The slope of the median from B to M_A is:

Slope of median from B to M_A = (b/2 - b) / (0 - 0) = -b/2

Since this median lies on the line y = mx + 2, we can set the slopes equal to find the relationship between b and m:

m = -b/2

Now, let's analyze the median from A to M_B. The slope of this median is:

Slope of median from A to M_B = (0 - 0) / (a/2 - a) = 0

Since this median is horizontal, it does not provide any constraints on m. Now, we need to find the values of m that satisfy the condition of the triangle's medians. We can express b in terms of m:

From m = -b/2, we have b = -2m.

Now, we can substitute this back into the triangle's coordinates. The triangle's area must be positive, which means both a and b must be positive. Therefore, we have:

-2m > 0, which implies m < 0.

Now, we can evaluate the options given:

• a. -12 (valid, since -12 < 0)
• b. 3/4 (invalid, since 3/4 > 0)
• c. 4/3 (invalid, since 4/3 > 0)
• d. 1/12 (invalid, since 1/12 > 0)

Thus, the only value of m for which such a triangle is possible is:

This means that the triangle can be constructed with the specified conditions when m equals -12.