IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

payal_bon

The coordinates of two vertices of a triangle are (-2,3)and (5,-1). If the orthocentre of the triangle is at the origin find the coordinates of its third vertex.

Ans:(-4,-7)

NOTE: Orthocentre is the point where the altitudes from the three vertices meet.

rooney

Let third vertex be(x,y)

Clearly the 2 altitudes are ~
L1> Line passing thru (-2,3) and origin
L2>Line passing thru (5,-1) and origin

Now, L1 will be perpendicular to line passing thru (5,-1) and (x,y)
So, (-3/2) x (y+1)/(x-5) = -1

//ly, L2 will be perpendicular to line passing thru (-2,3) and (x,y)
So, (-1/5) x (y-3)/(x+2) = -1

Solving these 2 eqs, u get answer as (-4,-7)

payal_bon

Just One Thing. I couldn't understand how you formed the equation for l1 and l2. Please do explain!!!

payal_bon

Just One thing. I couldn't understand how you formed the equations for L1 and L2. Please do Explain!!!

rooney

See. L1 passes through (-2,3) and (0,0). So slope is 3/-2. Similarly you get slope of L2.
Slope of side of triangle perpendicular to L1 is (y+1)/(x-5) as it passes through (x,y) and (5,-1). Similarly you get slope of the slope of the line perpedicular to the other altitude.

Since they are perpendicular, product of their slopes is -1.

Thats how we get the 2 equations

-Rohil

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