Question

For how many integral values of m do the lines

y + mx – 1 = 0 and 3x + 4y = 9 intersect in points having integral

coordinates –

• A. 0
• B. 1
• C. 2
• D. Infinite

Answer 1

Answer: (C)

2

Answer 2

We have 3x + 4y = 9

i.e x = $\frac{9 - 4y}{3}$ = 3 – $\frac{4y}{3}$

Thus, points lying on the above line and having integral coordinates are given by

P ŗ (3 – 4k, 3k) k Ī I

If P also lies on y + mx – 1 = 0, then we have

3k + m (3 – 4k) – 1 = 0

gives m = $\frac{3k - 1}{4k - 3}$, k Ī I

For m to be an integer, we have

|4k – 3| £ |3k – 1|

i.e. – (3k – 1) £ (4k – 3) £ (3k – 1)

i.e. $\frac{4}{7}$ £ k £ 2

There are only two integral values of k lying in the above integral, viz. k = 1, 2, Hence, there are only two integral values of m.