A straight line passes through the origin and is perpendicular to the line 2x + 3y = 6. This perpendicular line intersects 2x + 3y = 6 at point M. Using the concept of slopes and perpendicular lines, find the coordinates of the intersection point M.

Introduction / Context:
This question involves basic analytic geometry: slopes of lines, perpendicularity, and intersection points. Many competitive exams use such problems to test your ability to convert between different forms of line equations and to apply algebraic methods to find coordinates of intersection points.

Given Data / Assumptions:

• The given line has equation 2x + 3y = 6.
• A new line passes through the origin (0, 0).
• The new line is perpendicular to 2x + 3y = 6.
• The new line intersects 2x + 3y = 6 at point M, whose coordinates we need.

Concept / Approach:
First, find the slope of the given line by rewriting it in slope intercept form y = mx + c. A line perpendicular to a given line has a slope equal to the negative reciprocal of the original slope. Once we know the slope of the perpendicular line and the fact that it passes through the origin, we can write its equation. Finally, we solve the system of two linear equations in two variables to find the intersection point M.

Step-by-Step Solution:
Rewrite 2x + 3y = 6 as y = (-2/3)x + 2, so the slope of this line is -2/3.The slope of any line perpendicular to this is the negative reciprocal, which is 3/2.Since the perpendicular line passes through the origin, its equation is y = (3/2)x.To find M, solve 2x + 3y = 6 together with y = (3/2)x. Substitute y into the first equation: 2x + 3 * (3/2)x = 6.This gives 2x + 9x/2 = 6, so (4x + 9x) / 2 = 6, which simplifies to 13x / 2 = 6 and then x = 12/13. Substituting back into y = (3/2)x gives y = (3/2) * 12/13 = 18/13. So M is (12/13, 18/13).

Verification / Alternative check:
We can verify by showing that M satisfies both equations. For 2x + 3y = 6, substitute x = 12/13, y = 18/13: 2 * 12/13 + 3 * 18/13 = 24/13 + 54/13 = 78/13 = 6, which is correct. For y = (3/2)x, we have (3/2) * 12/13 = 36/26 = 18/13, which matches y. Also, the product of slopes of the two lines is (-2/3) * (3/2) = -1, confirming perpendicularity.

Why Other Options Are Wrong:

• (6/11, 9/11) and (6/7, 9/7) do not satisfy the equation 2x + 3y = 6.
• (-6/7, 9/7) does not lie on the line through the origin with slope 3/2 and also fails the first equation.
• (12/11, 18/11) has the same ratio of y to x but gives 2x + 3y not equal to 6.

Common Pitfalls:

• Using the same slope instead of the negative reciprocal when constructing a perpendicular line.
• Making arithmetic mistakes when solving the simultaneous equations.
• Forgetting that the perpendicular line must pass through the origin, which fixes its intercept as zero.

Final Answer:
(12/13, 18/13)