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प्रश्न
Find the co-ordinates of the orthocenter of the triangle whose vertices are A(3, –2), B(7, 6), C(–1, 2).
उत्तर
Let O be the orthocentre of ΔABC.
Let AD and BE be the altitudes on the sides BC and AC respectively.
Slope of side BC = `(2 - 6)/(-1 - 7) = (-4)/(-8) = 1/2`
∴ Slope of AD = – 2 ...[∵ AD ⊥ BC]
∴ Equation of line AD is
y – (– 2) = (– 2) (x – 3)
∴ y + 2 = – 2x + 6
∴ 2x + y – 4 = 0 .....(i)
Slope of side AC = `(-2 - 2)/(3 - (-1)) = (-4)/4` = – 1
∴ Slope of BE = 1 ...[∵ BE ⊥ AC]
∴ Equation of line BE is
y – 6 = 1(x – 7)
∴ y – 6 = x – 7
∴ x = y + 1 .....(ii)
Substituting x = y + 1 in (i), we get
2(y + 1) + y – 4 = 0
∴ 2y + 2 + y – 4 = 0
∴ 3y – 2 = 0
∴ y = `2/3`
Substituting y = `2/3` in (ii), we get
x = `2/3 + 1 = 5/3`
∴ Co-ordinates of orthocentre, O = `(5/3, 2/3)`
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