Question

If the distance between the points \( (2, -1) \) and \( (k, 3) \) is 5, then the possible values of \( k \) are:

• A. \( 2 \) and \( 6 \)
• B. \( -1 \) and \( 5 \)
• C. \( 1 \) and \( 3 \)
• D. \( 0 \) and \( 4 \)

Hint:

Tip: Always apply the distance formula carefully and square both sides to eliminate the root before solving.

Answer 1

The Correct Option is B

The problem involves finding the value of \( k \) given that the distance between the points \( (2, -1) \) and \( (k, 3) \) is 5. We use the distance formula which is given by:

\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Substituting the given coordinates, the formula becomes:

\( 5 = \sqrt{(k - 2)^2 + (3 + 1)^2} \)

Simplify inside the square root:

\( 5 = \sqrt{(k - 2)^2 + 16} \)

Square both sides to get rid of the square root:

\( 25 = (k - 2)^2 + 16 \)

Subtract 16 from both sides:

\( 9 = (k - 2)^2 \)

Take the square root of both sides:

\( k - 2 = \pm 3 \)

This gives us two equations:

\( k - 2 = 3 \) or \( k - 2 = -3 \)

Solve for \( k \) in each case:

• \( k = 5 \)
• \( k = -1 \)

Therefore, the possible values of \( k \) are \( -1 \) and \( 5 \).

Answer 2

Step 1: Use the distance formula.
Distance \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) Given: \[ \sqrt{(k - 2)^2 + (3 + 1)^2} = 5 \]

Step 2: Simplify the equation. 
\[ \sqrt{(k - 2)^2 + 16} = 5 \Rightarrow (k - 2)^2 + 16 = 25 \Rightarrow (k - 2)^2 = 9 \]

Step 3: Solve the quadratic. 
\[ k - 2 = \pm 3 \Rightarrow k = 5 \text{ or } -1 \] \[ (k - 2)^2 = 9 \Rightarrow k - 2 = \pm 3 \Rightarrow k = 2 + 3 = 5 \text{ or } 2 - 3 = -1 \] So correct values are \( k = 5 \) and \( k = -1 \)