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

Determine the value of \( k \) given that the distance between points \( (2, -1) \) and \( (k, 3) \) is 5. The distance formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the given points yields \( 5 = \sqrt{(k - 2)^2 + (3 - (-1))^2} \).

Simplify the expression within the square root: \( 5 = \sqrt{(k - 2)^2 + (3 + 1)^2} \), which becomes \( 5 = \sqrt{(k - 2)^2 + 4^2} \) and then \( 5 = \sqrt{(k - 2)^2 + 16} \).

Square both sides of the equation to eliminate the square root: \( 5^2 = (k - 2)^2 + 16 \), resulting in \( 25 = (k - 2)^2 + 16 \).

Isolate the squared term by subtracting 16 from both sides: \( 25 - 16 = (k - 2)^2 \), yielding \( 9 = (k - 2)^2 \).

Take the square root of both sides: \( \sqrt{9} = \sqrt{(k - 2)^2} \), which gives \( \pm 3 = k - 2 \).

This results in two separate equations:

• \( k - 2 = 3 \)
• \( k - 2 = -3 \)

Solving for \( k \) in each equation:

• \( k = 3 + 2 \Rightarrow k = 5 \)
• \( k = -3 + 2 \Rightarrow k = -1 \)

The possible values for \( k \) are \( -1 \) and \( 5 \).