JEE Coordinate (6)!

Geometry Level 4

The slopes of three sides of a triangle are -1 , -2 and 3 .

If the orthocenter of this triangle is the origin, and the locus of the centroid of this triangle can be expressed as $\dfrac xy = \dfrac mn$ , where $m$ and $n$ are coprime positive integers, find $m-n$ .

This is a part of my set Practice for JEE 2017!

2 7 3 6 1 5 8 4

1 solution

Sumanth R Hegde
Jan 11, 2017

Consider the vertices of the triangle to be

$(x_1, y_1) ,(x_2,y_2) , (x_3,y_3 )$

Now the slopes of the sides are known.Using this we will get a few equations in the unknowns .

Now , altitude drawn from a vertex passes through the origin.Using this we will get a few more equations.Solve these to express all variables in terms of only one variable.

Co ordinates of centroid are $( \frac{x_1 + x_2 + x_3}{3} , \frac{y_1 + y _2 + y_3}{3})$ . Take the ratio of $x$ and $y$ coordinates .

We will get $\large \color{#D61F06}{\frac {m}{n} = \frac{9}{2}}$

The answer is $\color{#3D99F6}{\boxed{7}}$

JEE Coordinate (6)!

This is a part of my set Practice for JEE 2017!

1 solution

0 pending reports