If the lines ` ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0` are concurrent then prove that,
`(1)/( 1-a) + (1) /( 1-b) + (1) /( 1-c) =`.

The lines ax + 2y + 1 = 0, bx + 3y +1 =0 and cx + 4y + 1=0 are concurrent if a, b, c are in A.P.

If the lines ax+y+1=0, x+by+1=0 and x+y+c=0 (a,b and c being distinct and different from 1) are concurrent the value of (a)/(a-1)+(b)/(b-1)+(c)/(c-1) is

If the points A(x,y), B(a,0) and C(0,b) are collinear, prove that (x)/(a) + (y)/(b) = 1

Prove that int_(0)^(1)log((1)/(x)-1)dx=0 .

If x sqrt(1 + y) + ysqrt(1 + x)= 0 , for -1 lt x lt 1 , prove that (dy)/(dx) = - (-1)/((1 + x)^(2))

If three lines whose equations are y=m_(1) x+c_(1), y=m_(2) x + c_(2) and y=m_(3) x + c_(3) are concurrent, then show that m_(1) (c_(2) - c_(3))+m_(2) (c_(3) - c_(1) ) + m_(3) ( c_(1) - c_(2) ) =0 .

Show that two lines a_(1) x + b_(14) y + c_(1) = 0 and a_(2) x + b_(2) y + c_(2) = 0 , where b_(1) , b_(2) ne 0 are (i) Parallel if (a_1)/( b_1) = (a_2)/( b_2) and

If x, y, z are not all zero & if ax + by + cz=0, bx+ cy + az=0 & cx + ay + bz = 0 , then prove that x: y : z = 1 : 1 : 1 OR 1 :omega:omega^2 OR 1:omega^2:omega , where omega is one ofthe complex cube root of unity.

If y= sin (pt), x= sin t , then prove that (1- x^(2))y_(2)-xy_(1)+ p^(2)y= 0