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JEE Main 202315 AprilMorning Shift
Exam
JEE Main
Year
2023
Date
15 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Straight Lines

JEE Main 2023 · 15 April · Morning Shift · Mathematics · Straight Lines

If (α, β) is the orthocenter of the triangle ABC with vertices A...

If (α,β)\left(\alpha, \beta\right) is the orthocenter of the triangle ABCABC with vertices A \left(3, – 7\right), B \left(– 1, 2\right) and C \left(4, 5\right), then 9α−6β+609\alpha - 6\beta + 60 is equal toIf (α, β) is the orthocenter of the triangle ABC with vertices A (3, – 7), B (– 1, 2) and C (4, 5), then 9α - 6β + 60

Options

  1. A2525
  2. B3535
  3. C3030
  4. D4040

Correct answer

A. 2525

Step-by-step solution

The required diagram will be:

If (α, β) is the orthocenter of the triangle ABC with vertices A (3, – 7), B (– 1, 2) and C (4, 5), then 9α - 6β + 60

Equation of AD:y+7=−53(x−3)</p><p>AD: y + 7 = \dfrac{- 5}{3}\left(x - 3\right)</p><p>⇒ 3y + 21 = - 5x + 15</p><p></p><p>⇒ 5x + 3y + 6 = 0 . . . . . . \left(i\right)</p><p>Equationof</p><p>Equation ofBE: y - 2 = \dfrac{- 1}{12}\left(x + 1\right)

⇒12y−24=–x−1⇒ 12y - 24 = – x - 1

⇒x=23−12y....(ii)⇒ x = 23 - 12y . . . . ( ii )

by \left(ii\right) 115 - 60y + 3y + 6 = 0</p><p></p><p>⇒ 57y = 121</p><p></p><p>y = \dfrac{121}{57},x=23−12×x = 23 - 12 \times\dfrac{121}{57}</p><p></p><p>∴ 9\alpha - 6\beta + 60 = 9 \times 23 - 108 \times \dfrac{121}{57} -6×6 \times\dfrac{121}{57} +6060

=207−242+60=25= 207 - 242 + 60 = 25

Hence this is the required option.

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