Question

The point on the line \(y = x\) equidistant from \((4,0)\) and \((5,1)\) is

• A. (2,2)
• B. (3,3)
• C. \((5/2,5/2)\)
• D. \((1/2,1/2)\)

Hint:

Use symmetry when point lies on line \(y=x\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A point on the line \( y = x \) can be represented as \( P(a, a) \). Being equidistant means the distance from P to the two given points must be the same.
: Key Formula or Approach:
Distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Step 2: Detailed Explanation:
Let the required point be \( P(a, a) \).
Distance from \( (4,0) \): \( d_1^2 = (a - 4)^2 + (a - 0)^2 \).
Distance from \( (5,1) \): \( d_2^2 = (a - 5)^2 + (a - 1)^2 \).
Since \( d_1 = d_2 \), we set \( d_1^2 = d_2^2 \): \[ (a - 4)^2 + a^2 = (a - 5)^2 + (a - 1)^2 \] Expand the terms: \[ a^2 - 8a + 16 + a^2 = a^2 - 10a + 25 + a^2 - 2a + 1 \] Combine like terms: \[ 2a^2 - 8a + 16 = 2a^2 - 12a + 26 \] Eliminate \( 2a^2 \) from both sides: \[ -8a + 16 = -12a + 26 \] \[ 12a - 8a = 26 - 16 \] \[ 4a = 10 \implies a = \frac{10}{4} = \frac{5}{2} \] The point is \( (5/2, 5/2) \).
Step 3: Final Answer:
The point is (5/2, 5/2).