Find the vertexof the triangle by finding intersection points on three lines

Remember that the orthocentre of a right angled triangle is the vertex containing the right angle. Here u can see that the lines x=3y and y= - 3x are mutually perpendicular bcoz the product of their slopes is = (1/3).(-3) = -1 .

Therefore the three lines form a right angled triangle.

Both the lines, x=3y and y= -3x  pass through the origin and hence their intersection point is the origin. Therefore the orthocentre of the triangle formed by the given st. lines is the origin. Now , the st. line 3x-4y=0 or 3x=4y also passes through the origin and hence it passes through the orthocentre of the given triangle .

You may also keep it in mind that the circumcentre of a right angled triangle is the mid point of the hypotenuse .

Question is incomplete ....i guess ?

 Ques: the equation to the sides of a triangle are x-3y =0 ,, 4x+3y=5 and 3x+y =0. the line 3x-4y =? pases through..A)the incentrre b) the orthocentre c)the centroid d) the circumcentre 

Incentre :  Point where all the angle bisector intersect

Orthocentre: Point where all the altitude of the triangle intersect.

Circumcentre: Point where all the perpendicular bisector of the triangle intersect.

Centroid : The point where all the medians intersect ....

Three sides     x - 3y = 0       4x +3y = 5      3x + y = 0  

In this case  the two lines are perpendicular to each other....that is   3x + y = 0 and x-3y = 0 ...

Hence the triangle is a right angle triangle with  hypotenous   4x + 3y = 5

Now in question you have given the line to which we have to choose the option is

 3x-4y = K(i  suppose)

Now  3x-4y = K is perpendicular to  hypotenous .......since it is perpendicular ....so it can pass through  orthocentre.....or circumcentre....

Answer will depend on the value of  K....

How ? 

You will solve  x-3y = 0 and  3x +y = 0 ...i.e (0,0) ...if (0,0) satisfy the 3x - 4y = k then it will be orthocentre.....otherwise orthocentre is not the correct option...

Also ...if it will be circumcentre only if ...it passes through middle point of hypotenous.....

Hence K should be given in the question ..definately