First slide

Equation of line in Straight lines

Question

If two vertices of a triangle are (-2, 3) and (5, -l), the orthocenter lies at the origin, and the centroid on the line $x + y = 7$ , then the third vertex lies at

Moderate

A

(7,4)
B

(8, 14)
C

(12,21)
D

None of these

Solution

$S u p p o s e t h a t t h e t h i r d v e r t e x i s A (h, k) B C i s p e r p e n d i c u l a r t o A O b e c a u s e O i s o r t h o c e n t e r H e n c e, \frac{k}{h} \times \frac{- 4}{7} = - 1 4 h - 7 k = 0 T h e c e n t r o i d o f t h e t r i a n g l e l i e s o n x + y = 7 (\frac{- 2 + 5 + h}{3}, \frac{3 - 1 + k}{3}) l i e s o n x + y = 7 3 + h + 2 + k = 21 h + k = 16 s o l v i n g t h e a b o v e t w o e q u a t i o n s 7 k + 4 k = 64 11 k = 64 k = \frac{64}{11}, h = \frac{112}{11} T h i r d v e r t e x i s (\frac{64}{11}, \frac{112}{11})$

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