First slide

Cartesian plane

Question

If two vertices of a triangle are (-2,3) and (5,-1), the orthocentre 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

Given O(0,0) is the orthocentre. Let A(h,k) be the third vertex, and B(-2,3) and C(5,-1) the other two vertices. Then the slope of theline throught A and O is k/h, while the line through B and C has the slope -4/7. By the property of the orthocentre, these two lines must be perependicular. So, we have
$(\frac{h}{k}) (- \frac{4}{7}) = - 1 or \frac{k}{h} = \frac{7}{4}(i)$
Also, $\frac{5 - 2 + h}{3} + \frac{- 1 + 3 + k}{3} = 7$
or $h + k = 16(ii)$
Which is not satisfied by the points given in (i),(ii) or (iii).

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