Finding Vertex B in a Triangle Given Coordinates and Orthocentre

[utkarshakash]

Homework Statement

A triangle ABC is given where vertex A is (1,1) and the orthocentre is (2,4). Also sides AB and BC are members of the family of lines ax+by+c=0 where a,b,c are in Arithmetic Progression.

Find vertex B.

Homework Equations

The Attempt at a Solution

I can write the equation of AO (O being orthocentre). But I need to somehow get vertex B and I think I need to have some more equations to do that. Can someone explain me how to apply the second hint given in the question?

 

[Saitama]

Homework Statement

A triangle ABC is given where vertex A is (1,1) and the orthocentre is (2,4). Also sides AB and BC are members of the family of lines ax+by+c=0 where a,b,c are in A.P.

Find vertex B.

Homework Equations

The Attempt at a Solution

I can write the equation of AO (O being orthocentre). But I need to somehow get vertex B and I think I need to have some more equations to do that. Can someone explain me how to apply the second hint given in the question?

I haven't solved the question but this should give you some idea. You can find the equation of line for BC (which is perpendicular to AO) using the fact that a,b,c are in A.P. Assume that vertex B is (e,f). This point satisfies the line BC and you get a relation between e and f. Now find the line AB and again use the fact that a,b and c are in A.P.

PS: You should define what A.P means here. I take it as arithmetic progression but not everyone here uses the abbreviation A.P for arithmetic progression. :)