Question

If the lines y = x + 3 and ax + y = 5 intersect at integral co-ordinates in first quadrant, then number of integral values of a is:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

We are given the lines

$$y = x + 3$$

and

$$ax + y = 5.$$

Substitute $y = x + 3$ into $ax + y = 5$:

$$ax + (x+3) = 5 \quad \Rightarrow \quad (a+1)x + 3 = 5.$$

Subtract 3 from both sides:

$$(a+1)x = 2 \quad \Rightarrow \quad x = \frac{2}{a+1}.$$

For the intersection $(x, y)$ to be in the first quadrant with integral coordinates, $x$ must be a positive integer. Therefore, $a+1$ must be a positive divisor of 2. The positive divisors of 2 are 1 and 2.

• If $a+1 = 1$, then $a = 0$ and $x = \frac{2}{1} = 2$. Then $y = 2 + 3 = 5$.
• If $a+1 = 2$, then $a = 1$ and $x = \frac{2}{2} = 1$. Then $y = 1 + 3 = 4$.

Both $(2,5)$ and $(1,4)$ lie in the first quadrant with integer coordinates.

Thus, there are 2 integral values of $a$.