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

Question

Infinity Learn · Accepted Answer

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 (hk)($$−$$47)=$$−$$1$$  or $$kh=74________(i)Also$$, $$5$$−$$2+h3+$$−$$1+3+k3=7or$$ $$h+k=16________(ii)Which$$ is not satisfied by the points given in $$(i),(ii) or (iii).

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

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