If Q(0, 1) is equidistant from point P (5, -3) and R (x, 6), then find the value of x

यदि Q(0,1) बिन्दुओं P(5,-3) और R (x,6) से समदूरस्थ हैं, तो x के मान ज्ञात कीजिए।

Options

A. `pm 2`
B. `pm 4`
C. `pm 6`
D. `pm 5`

Explanation

To solve the problem where point Q(0, 1) is equidistant from points P(5, -3) and R(x, 6), we can follow these steps:

### Step 1: Calculate the distance from Q to P
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points Q(0, 1) and P(5, -3):
- \( x_1 = 0, y_1 = 1 \)
- \( x_2 = 5, y_2 = -3 \)

Using the distance formula:
\[ d(Q, P) = \sqrt{(5 - 0)^2 + (-3 - 1)^2} \]
Calculating this gives:
\[ d(Q, P) = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \]

### Step 2: Calculate the distance from Q to R
Now, we calculate the distance from Q(0, 1) to R(x, 6):
\[ d(Q, R) = \sqrt{(x - 0)^2 + (6 - 1)^2} \]
This simplifies to:
\[ d(Q, R) = \sqrt{x^2 + 5^2} = \sqrt{x^2 + 25} \]

### Step 3: Set the distances equal
Since Q is equidistant from P and R, we set the two distances equal:
\[ \sqrt{41} = \sqrt{x^2 + 25} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ 41 = x^2 + 25 \]

### Step 5: Solve for x
Rearranging the equation:
\[ x^2 = 41 - 25 \]
\[ x^2 = 16 \]
Taking the square root of both sides:
\[ x = \pm 4 \]

### Final Answer
Thus, the values of \( x \) are:
\[ x = 4 \quad \text{or} \quad x = -4 \]

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