Find the values of y for which the distance between the points `P(2,\ 3)` and `Q(10 ,\ y)`is 10 units.

Options

A. `y=9`
B. `y=-5`
C. `y=-3`
D. Both A and C

Explanation

To find the values of \( y \) for which the distance between the points \( P(2, 3) \) and \( Q(10, y) \) is 10 units, we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, \( P(2, 3) \) and \( Q(10, y) \).

### Step 2: Substitute the Coordinates into the Distance Formula
Substituting the coordinates of points \( P \) and \( Q \) into the distance formula gives:
\[ 10 = \sqrt{(10 - 2)^2 + (y - 3)^2} \]

### Step 3: Simplify the Equation
Calculating \( (10 - 2)^2 \):
\[ 10 - 2 = 8 \quad \Rightarrow \quad 8^2 = 64 \]
So, we have:
\[ 10 = \sqrt{64 + (y - 3)^2} \]

### Step 4: Square Both Sides
To eliminate the square root, square both sides:
\[ 10^2 = 64 + (y - 3)^2 \]
\[ 100 = 64 + (y - 3)^2 \]

### Step 5: Isolate the \( (y - 3)^2 \) Term
Subtract 64 from both sides:
\[ 100 - 64 = (y - 3)^2 \]
\[ 36 = (y - 3)^2 \]

### Step 6: Solve for \( y - 3 \)
Take the square root of both sides:
\[ y - 3 = \pm 6 \]

### Step 7: Find the Values of \( y \)
Now, solve for \( y \):
1. \( y - 3 = 6 \) leads to \( y = 9 \)
2. \( y - 3 = -6 \) leads to \( y = -3 \)

### Final Answer
The values of \( y \) for which the distance between points \( P(2, 3) \) and \( Q(10, y) \) is 10 units are:
\[ y = 9 \quad \text{and} \quad y = -3 \]