Two vertices of a rectangle lie on the line Y = 2X + λ and coordinates of the…
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Two vertices of a rectangle lie on the line Y = 2X + λ and coordinates of the rest two vertices are diagonally opposite to each other are (1,3) and (5,1). Find the value of λ.
- A.
4
- B.
5
- C.
-4
- D.
-3
Attempted by 4 students.
Show answer & explanation
Correct answer: C
Key idea: Diagonals of a rectangle bisect each other, so the midpoint of one diagonal equals the midpoint of the other and must lie on the given line.
Step 1: Find the midpoint of the given opposite vertices (1, 3) and (5, 1):
Midpoint = ((1+5)/2, (3+1)/2) = (3, 2).
Step 2: This midpoint (3, 2) lies on the line y = 2x + λ, so substitute x = 3 and y = 2.
2 = 2·3 + λ ⇒ 2 = 6 + λ.
Step 3: Solve for λ:
λ = 2 − 6 = −4.
Answer: −4