Q. 134.3( 6 Votes )
Find the coordinates of the orthocenter of the triangle whose vertices are ( - 1, 3), (2, - 1) and (0, 0).
Answer :
Given: coordinates of the orthocenter of the triangle whose vertices are ( - 1, 3), (2, - 1) and (0, 0).
Assuming:
A (0, 0), B (−1, 3) and C (2, −1) be the vertices of the triangle ABC.
Let AD and BE be the altitudes.
To find:
Orthocenter of the triangle.
Explanation:
AD⊥BC and BE⊥AC
∴ The slope of AD × Slope of BC = −1
The slope of BE × Slope of AC = −1
Here, the slope of BC = 
and slope of AC = 
∴ slope of AD × ( - 4/3) = - 1 and slope of BE × ( - 1/2) = - 1
⇒ slope of AD 
and slope of BE = 2
The equation of the altitude AD passing through A (0, 0) and having slope 
is
y - 0
( x - 0)
⇒ y 
x …..(1)
The equation of the altitude BE passing through B (−1, 3) and having slope 2 is
y - 3 = 2(x + 1)
⇒ 2x – y + 5 = 0 …….(2)
Solving (1) and (2):
x = − 4, y = − 3
Hence, the coordinates of the orthocentre is (−4, −3).
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