If the distance between the points (3,6) and (p,10) is `2sqrt5`, then the value of p is :

Options

A. 4
B. `5 `
C. `-4`
D. 0

Answer

To find the value of \( p \) given that the distance between the points \( (3, 6) \) and \( (p, 10) \) is \( 2\sqrt{5} \), we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (3, 6) \) and \( (x_2, y_2) = (p, 10) \).

### Step 2: Substitute the Points into the Formula
Substituting the coordinates into the distance formula, we have:
\[ 2\sqrt{5} = \sqrt{(p - 3)^2 + (10 - 6)^2} \]

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ 2\sqrt{5} = \sqrt{(p - 3)^2 + 4^2} \]
This simplifies to:
\[ 2\sqrt{5} = \sqrt{(p - 3)^2 + 16} \]

### Step 4: Square Both Sides
To eliminate the square root, square both sides:
\[ (2\sqrt{5})^2 = (p - 3)^2 + 16 \]
This gives:
\[ 4 \cdot 5 = (p - 3)^2 + 16 \]
\[ 20 = (p - 3)^2 + 16 \]

### Step 5: Isolate the Square Term
Subtract 16 from both sides:
\[ 20 - 16 = (p - 3)^2 \]
\[ 4 = (p - 3)^2 \]

### Step 6: Take the Square Root
Now, take the square root of both sides:
\[ \sqrt{4} = p - 3 \]
This gives us:
\[ 2 = p - 3 \quad \text{or} \quad -2 = p - 3 \]

### Step 7: Solve for \( p \)
Now, solve for \( p \) in both cases:
1. \( p - 3 = 2 \) leads to \( p = 5 \)
2. \( p - 3 = -2 \) leads to \( p = 1 \)

### Final Answer
Thus, the possible values of \( p \) are \( 5 \) and \( 1 \).

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