If the distance between P (4 , 0) and Q (0, x) is 5 units , the value of x will be

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the value of \( x \) such that the distance between the points \( P(4, 0) \) and \( Q(0, x) \) is 5 units, we can follow these steps:

### Step 1: Write down the distance formula
The distance \( D \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 2: Substitute the coordinates of points P and Q
In our case, the coordinates of point \( P \) are \( (4, 0) \) and the coordinates of point \( Q \) are \( (0, x) \). Therefore, we can substitute these values into the distance formula:
\[ D = \sqrt{(0 - 4)^2 + (x - 0)^2} \]

### Step 3: Simplify the expression
Now, simplify the expression:
\[ D = \sqrt{(-4)^2 + x^2} = \sqrt{16 + x^2} \]

### Step 4: Set the distance equal to 5
According to the problem, the distance \( D \) is equal to 5 units:
\[ \sqrt{16 + x^2} = 5 \]

### Step 5: Square both sides to eliminate the square root
To eliminate the square root, we square both sides of the equation:
\[ 16 + x^2 = 25 \]

### Step 6: Solve for \( x^2 \)
Now, isolate \( x^2 \) by subtracting 16 from both sides:
\[ x^2 = 25 - 16 \]
\[ x^2 = 9 \]

### Step 7: Find the values of \( x \)
To find \( x \), take the square root of both sides:
\[ x = \pm 3 \]

### Final Answer
Thus, the values of \( x \) are \( 3 \) and \( -3 \).

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