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JEE Main 202623 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Straight Lines

JEE Main 2026 · 23 January · Evening Shift · Mathematics · Straight Lines

Let A(1,2) and C(-3,-6) be two diagonally opposite vertices of a...

Let A(1,2)\mathrm{A}(1,2) and C(−3,−6)\mathrm{C}(-3,-6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line 7x−y=147 x-y=14. If B(α,β)\mathrm{B}(\alpha, \beta) and D(γ,δ)\mathrm{D}(\gamma, \delta) are the other two vertices, then ∣α+β+γ+δ∣|\alpha+\beta+\gamma+\delta| is equal to

Options

  1. A6
  2. B9
  3. C3
  4. D1

Correct answer

A. 6

Step-by-step solution

Midpoint of diagonal ACAC is M=(−1,−2)M = (-1,-2). Slope of AC=−6−2−3−1=2AC = \frac{-6-2}{-3-1} = 2.
In a rhombus, diagonals are perpendicular bisectors. So BDBD has slope −12-\frac{1}{2} and passes through MM: y+2=−12(x+1)y + 2 = -\frac{1}{2}(x+1).
ADAD has slope 7: δ−2γ−1=7⇒δ=7γ−5\frac{\delta - 2}{\gamma - 1} = 7 \Rightarrow \delta = 7\gamma - 5.
DD lies on BDBD: δ=−γ2−52\delta = -\frac{\gamma}{2} - \frac{5}{2}.
Solving: 7γ−5=−γ2−52⇒γ=137\gamma - 5 = -\frac{\gamma}{2} - \frac{5}{2} \Rightarrow \gamma = \frac{1}{3}, δ=−83\delta = -\frac{8}{3}.
B=(2(−1)−13, 2(−2)+83)=(−73,−43)B = (2(-1)-\frac{1}{3},\, 2(-2)+\frac{8}{3}) = (-\frac{7}{3}, -\frac{4}{3}).
∣α+β+γ+δ∣=∣−73−43+13−83∣=∣−6∣=6|\alpha+\beta+\gamma+\delta| = \left|-\frac{7}{3}-\frac{4}{3}+\frac{1}{3}-\frac{8}{3}\right| = |-6| = 6.
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