Pair of equations $4x+py+8=0$ and $2x+2y+2=0$ have unique solution, if

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Accepted Answer

To determine when the pair of equations $4x + py + 8 = 0$ and $2x + 2y + 2 = 0$ have a unique solution, we need to consider the conditions for a system of linear equations to have a unique solution. A system of two linear equations in two variables has a unique solution if and only if the equations are not parallel, which means their slopes must be different.

Let's find the slopes of the given equations:

1. The first equation is $4x + py + 8 = 0$. We can rewrite it in the slope-intercept form $y = mx + c$ as follows:
   $$
   py = -4x - 8 \implies y = -\frac{4}{p}x - \frac{8}{p}
   $$
   Thus, the slope of the first equation is $-\frac{4}{p}$.

2. The second equation is $2x + 2y + 2 = 0$. Similarly, rewriting it in the slope-intercept form:
   $$
   2y = -2x - 2 \implies y = -x - 1
   $$
   The slope of the second equation is $-1$.

For the system to have a unique solution, the slopes must be different:
$$
-\frac{4}{p} \neq -1
$$

Solving this inequality:
$$
\frac{4}{p} \neq 1 \implies p \neq 4
$$

Thus, the condition for the system to have a unique solution is $p \neq 4$, which corresponds to Option 1.

**Explanation of Incorrect Options:**

- **Option 2: $p \ne 2$**  
  This option is incorrect because substituting $p = 2$ into the slope condition gives $-\frac{4}{2} = -2$, which is not equal to $-1$. Therefore, the equations would still have different slopes and a unique solution.

- **Option 3: $p \ne 1$**  
  This option is incorrect because substituting $p = 1$ into the slope condition gives $-\frac{4}{1} = -4$, which is not equal to $-1$. Thus, the equations would still have different slopes and a unique solution.

- **Option 4: $p \ne 0$**  
  This option is incorrect because substituting $p = 0$ would make the first equation undefined in terms of slope (as division by zero is not possible), but it does not affect the condition for uniqueness directly as long as $p$ is not 4.

Therefore, the correct answer is indeed Option 1: $p \ne 4$.