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Equation of Family of Circles Touching a Line and Passing through a Given Point on the Line

The triangle ...

Question

The triangle ABC has medians AD, BE, CF . AD lies along the line y = x + 3, BE lies along the line y = 2x + 4, AB has length 60 and angle C = 90°, then the area of triangle ABC is
(A) 400 (B) 200
(C) 100 (D) none of these

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Solution

The easiest way to solve this question is by shifting the centroid to origin.
Thus now equations of median are y=x and y=2x.
Now the coordiantes of A can be taken as (a,a) and that of B can be taken as (b,2b)
Using centroid formula C(-a-b,-a,-2b)
Equate hypotenuse length equal to 60 get one equation. Second equation comes from slope of AC and BC product is -1.Solving both the equations you will get ab=(-800/3).
Find out area of triangle using all three co-ordinates. Area=3/2 (ab)
=400

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