The length of line PQ is 10 units and the co-ordinates of P are (2, -3), calculate the co-ordinates of point Q, if its absicissa is 10.

Answer

To solve the problem step by step, we will use the distance formula and the given information.

### Step 1: Understand the Given Information
We know the following:
- The length of line segment PQ is 10 units.
- The coordinates of point P are (2, -3).
- The abscissa (x-coordinate) of point Q is 10.

### Step 2: Set Up the Coordinates of Point Q
Since the abscissa of point Q is given as 10, we can denote the coordinates of point Q as (10, y), where y is the unknown y-coordinate we need to find.

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

Substituting the coordinates into the distance formula:
\[ 10 = \sqrt{(10 - 2)^2 + (y - (-3))^2} \]

### Step 4: Simplify the Equation
Now, simplify the equation:
\[ 10 = \sqrt{(8)^2 + (y + 3)^2} \]
\[ 10 = \sqrt{64 + (y + 3)^2} \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ 100 = 64 + (y + 3)^2 \]

### Step 6: Isolate the Square Term
Now, isolate the square term:
\[ 100 - 64 = (y + 3)^2 \]
\[ 36 = (y + 3)^2 \]

### Step 7: Take the Square Root
Taking the square root of both sides gives us two possible equations:
\[ y + 3 = 6 \quad \text{or} \quad y + 3 = -6 \]

### Step 8: Solve for y
Now, solve for y in both cases:
1. \( y + 3 = 6 \)
\[ y = 6 - 3 = 3 \]
2. \( y + 3 = -6 \)
\[ y = -6 - 3 = -9 \]

### Step 9: Final Coordinates of Point Q
Thus, the coordinates of point Q can be:
- \( Q(10, 3) \)
- \( Q(10, -9) \)

### Conclusion
The final coordinates of point Q are \( (10, 3) \) and \( (10, -9) \).

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