Show that straight lines `(A^2-3B^2)x^2+8A Bx y+(B^2-3A^2)y^2=0` form with the line `A x+B y+C=0` an equilateral triangle of area `(C^2)/(sqrt(3(A^2+B^2)))` .

Show that the straight line x^2-4xy+y^2=0 and x + y = 3 form an equilateral triangle.

Show that the line x^(2)-4xy+y^(2)=0 and x+y=3 from an equilateral triangle.

Show that the straight line x-2y+3=0 and 6x+3y+8=0 are perpendicular .

Find the angle between the pair of straight lines given by (a^(2)-3b^(2))x^(2)+8ab xy+(b^(2)-3a^(2))y^(2)=0

Find the maximum area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which touches the line y=3x+2.

If the straight lines 2x+3y-1=0,x+2y-1=0,and ax+by-1=0 form a triangle with the origin as orthocentre, then (a , b) is given by

Show that the following points taken in order form an equilateral triangle in each case . (i) A(2,2), B(-2,-2), C-2sqrt(3),2sqrt(3)) (ii) A(sqrt(3),2),B(0,1),C(0,3)

Show that straight lines joining the origin to the points of intersection of 3x-2y+2=0 and 3x^2+5x-2y^2+4x+5y=0 are at right angles .

P is a point on the line y+2x=1, and Q and R two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these