For what value of k will the lines 2x + 7ky - 8 = 0 and x + y - 9 = 0 have no solution?

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Accepted Answer

For two lines to have no solution, they must be parallel but not coincident. The general form of a line is ax + by + c = 0. The lines are:

1) 2x + 7ky - 8 = 0
2) x + y - 9 = 0

For parallelism:
(a1/b1) = (a2/b2) and (c1/c2) ≠ (a1/a2) or (b1/b2)

Comparing:
First line: a1 = 2, b1 = 7k, c1 = -8
Second line: a2 = 1, b2 = 1, c2 = -9

For parallelism:
2/(7k) = 1/1 → 2 = 7k → k = 2/7

But we must also check that the lines are not coincident. For coincidence:
(a1/a2) = (b1/b2) = (c1/c2)

Check c1/c2:
-8/-9 = 8/9

So, for no solution, k = 2/7 and 8/9 ≠ 2/1 or 7k/1. So, k = 2/7 is the value for parallelism.

But let's check if this matches the options. Option 1 is 2/7, but let's check the coincidence condition:

(a1/a2) = 2/1 = 2
(b1/b2) = (7k)/1 = 7k
(c1/c2) = (-8)/(-9) = 8/9

For the lines to be coincident:
2 = 7k = 8/9 → 7k = 2 → k = 2/7 (already found)
But 7k = 8/9 → k = 8/63
So, for k = 2/7, the lines are parallel and not coincident.

Therefore, the correct answer is k = 2/7. However, the answer key marks 9/7. Let's check if that's a miscalculation.

Let's check the ratios:
(a1/a2) = 2/1 = 2
(b1/b2) = 7k/1 = 7k
For parallelism: 7k = 2 → k = 2/7

So, the correct answer is 2/7, not 9/7. But according to the answer key, 9/7 is marked. Let's check if there's a misinterpretation.

If we set (a1/a2) = (b1/b2):
2/1 = 7k/1 → 7k = 2 → k = 2/7

So, the correct answer is 2/7.

Thus, Option 1 is correct.

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