A straight line passes through the origin and is perpendicular to the line 3x − 2y = 6. This perpendicular line intersects 3x − 2y = 6 at point M. Find the exact coordinates of point M.

Introduction / Context:
This coordinate geometry question involves slopes of perpendicular lines and their point of intersection. You are given a line in standard form and asked to find the point where it is intersected by another line that passes through the origin and is perpendicular to it. Problems of this type are common in analytic geometry and test understanding of slopes, perpendicularity, and solving simultaneous linear equations.

Given Data / Assumptions:

• Line L1 has equation 3x − 2y = 6.
• Line L2 passes through the origin (0, 0).
• Line L2 is perpendicular to L1.
• The lines intersect at point M, whose coordinates we must find.
• All coordinates are real numbers.

Concept / Approach:
First, rewrite the given line 3x − 2y = 6 in slope intercept form to find its slope. The slope of a line perpendicular to this line is the negative reciprocal of that slope. Using the fact that L2 passes through the origin, we can write its equation immediately. Then we solve the system of equations formed by L1 and L2 to find their intersection point M. This approach combines slope concepts with solving linear systems.

Step-by-Step Solution:
Step 1: Rewrite L1: 3x − 2y = 6. Solve for y: −2y = 6 − 3x, so y = (3/2)x − 3.Step 2: The slope of L1 is 3/2.Step 3: The slope of a line perpendicular to L1 is the negative reciprocal, which is −2/3.Step 4: Since L2 passes through the origin, its equation is y = (−2/3)x.Step 5: To find M, solve the system consisting of y = (3/2)x − 3 and y = (−2/3)x.Step 6: Set the right sides equal: (3/2)x − 3 = (−2/3)x.Step 7: Add (2/3)x to both sides: (3/2x + 2/3x) − 3 = 0.Step 8: Find a common denominator for the coefficients: 3/2 = 9/6 and 2/3 = 4/6, so 3/2x + 2/3x = 13/6x.Step 9: The equation becomes 13/6 x − 3 = 0, so 13/6 x = 3 and x = 3 × 6/13 = 18/13.Step 10: Substitute x = 18/13 into y = (−2/3)x: y = (−2/3) × 18/13 = −36/39 = −12/13.Step 11: Thus M has coordinates (18/13, −12/13).

Verification / Alternative check:
Check that M lies on both lines. For line L1, substitute into 3x − 2y = 6: 3 × 18/13 − 2 × (−12/13) = 54/13 + 24/13 = 78/13 = 6, so M lies on L1. For line L2, check y = (−2/3)x: (−12/13) = (−2/3) × 18/13 = −36/39 = −12/13, so M lies on L2 as well. This confirms that the coordinates of M are correct.

Why Other Options Are Wrong:
The point (18/13, 12/13) has the wrong sign for y and does not satisfy y = (−2/3)x. The points (−18/13, −12/13) and (−18/13, 12/13) use incorrect signs for x and do not satisfy the equation 3x − 2y = 6. The origin (0, 0) lies on the perpendicular line but not on the given line, so it cannot be the intersection point. Only (18/13, −12/13) satisfies both line equations simultaneously.

Common Pitfalls:
Some learners confuse the slope of the perpendicular line and take it as the reciprocal rather than the negative reciprocal. Others make arithmetic errors when combining fractional slopes or solving for x. Carefully converting to slope intercept form, correctly determining the perpendicular slope, and working step by step with fractions helps avoid these mistakes.

Final Answer:
The coordinates of point M are (18/13, −12/13).