If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept equal to –4, then a value of k is :

Solution

Maths sir · 4 steps

Every point on a perpendicular bisector is equally far from P and Q, so use the point (0, −4) directly.

1. Use the intercept point. A y-intercept of -4 means the bisector passes through (0, -4). Any point on the perpendicular bisector of PQ is at the same distance from P and from Q.

   (0-1)² + (-4-4)² = (0-k)² + (-4-3)²

2. Solve for k. The left side is 1 + 64 = 65 and the right side is k² + 49. So k² = 16, which gives k = 4 or k = -4.
3. Cross-check. With midpoint ((1+k)/2, 7/2) and slope k-1, the bisector's y-intercept works out to (8-k²)/2. For k² = 16 this is -4, as required.
4. So. Of the two values, only k = -4 is among the options, so the answer is (b).

The idea to keep. For a perpendicular bisector, skip the slope work when you can: any point on it is equidistant from both ends, so equate the squared distances.

Answer from the NTA final answer key. Solution by PrepWiser.