If the line 2x − 3y = 11 is perpendicular to the line 3x + ky = −4, what is the value of k in coordinate geometry?

Introduction / Context:
This aptitude question tests your understanding of the relationship between the slopes of two perpendicular straight lines in coordinate geometry. When two lines are perpendicular, their slopes have a special product, and you can use that fact to find the unknown constant k in the second line. This is a common pattern in school level algebra and coordinate geometry exams.

Given Data / Assumptions:

- First line: 2x − 3y = 11
- Second line: 3x + ky = −4
- The two lines are perpendicular to each other
- x and y are real variables and k is a real constant

Concept / Approach:
For any straight line written in the form ax + by + c = 0, the slope is given by m = −a/b (provided b is not zero). Two lines with slopes m1 and m2 are perpendicular if and only if m1 * m2 = −1. Therefore, the key idea is to convert both line equations into slope form, identify their slopes, and then enforce the perpendicularity condition to solve for k.

Step-by-Step Solution:
Step 1: Rearrange the first line 2x − 3y = 11 into slope form y = mx + c. We get −3y = −2x + 11, so y = (2/3)x − 11/3. Thus the slope of the first line is m1 = 2/3.Step 2: Rearrange the second line 3x + ky = −4 into slope form. We get ky = −3x − 4, so y = (−3/k)x − 4/k. Therefore the slope of the second line is m2 = −3/k.Step 3: Use the perpendicularity condition m1 * m2 = −1. Substitute m1 = 2/3 and m2 = −3/k to get (2/3) * (−3/k) = −1.Step 4: Simplify the left side: (2/3) * (−3/k) = −2/k. So the equation becomes −2/k = −1.Step 5: Multiply both sides by k and divide by 1 to get −2 = −k, which gives k = 2.

Verification / Alternative check:
If k = 2, then the second line becomes 3x + 2y = −4. Its slope is m2 = −3/2. The product of the slopes is m1 * m2 = (2/3) * (−3/2) = −1, which confirms that the lines are indeed perpendicular. So k = 2 is consistent.

Why Other Options Are Wrong:
- For k = −2, the slope of the second line becomes 3/2 and the product of slopes is (2/3) * (3/2) = 1, so the lines are not perpendicular.
- For k = 1, the slope becomes −3 and the product is (2/3) * (−3) = −2, not −1.
- For k = −1, the slope becomes 3 and the product is (2/3) * 3 = 2, again not −1.
- The value 3 does not satisfy the perpendicularity condition either, since substituting k = 3 gives a slope of −1 and a product of (2/3) * (−1) = −2/3.

Common Pitfalls:
Students often forget that the standard condition for perpendicular lines is m1 * m2 = −1, not m1 = −m2. Another frequent error is in computing slopes from the general form ax + by + c = 0; remember that the slope is −a/b, not a/b. Also be careful while simplifying fractions so that negative signs are handled consistently.

Final Answer:
The value of k is 2.