If the lines whose equations are `y=m_1 x+ c_1 , y = m_2 x + c_2 and y=m_3 x + c_3 ` meet in a point, then prove that : `m_1 (c_2 - c_3) + m_2 (c_3 - c_1) + m_3 (c_1 - c_2) =0`

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

(i) Find the value of 'a' if the lines 3x-2y+8=0 , 2x+y+3=0 and ax+3y+11=0 are concurrent. (ii) If the lines y=m_(1)x+c_(1) , y=m_(2)x+c_(2) and y=m_(3)x+c_(3) meet at point then shown that : c_(1)(m_(2)-m_(3))+c_(2)(m_(3)-m_(1))+c_(3)(m_(1)-m_(2))=0

Find the coordinates that the straight lines y=m_(1)x+c_(1),y=m_(2)x+c_(2) and y=m_(2)x+c_(3) may meet in a point.

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0

The line y=m x+1 is a tangent to the curve y^2=4x , if the value of m is (a) 1 (b) 2 (c) 3 (d) 1/2

If y+3=m_1(x+2) and y+3=m_2(x+2) are two tangents to the parabola y_2=8x , then (a)m_1+m_2=0 (b) m_1+m_2=-1 (c)m_1+m_2=1 (d) none of these

The line y=mx+1 is a tangent to the curve y^2=4x if the value of m is (A) 1 (B) 2 (C) 3 (D) 1/2 .

31. If the lines represented byax? + 2hxy + by2 + 2gx + 2 fy+c= 0 are writtenin the form y=mx+c1 and y = m2x+C2,then1) m + m2 = alb, mm2 = 2h1b2) m + m2 =-2h/b, mm2 = alb3) m, + m2 = -2h/b, m,m, = -b/a4) m, + m2 = 2h/b, m,m, = a / b

X Y is drawn parallel to the base B C of a A B C cutting A B at X and A C at Y . If A B=4 B X and Y C=2c m , then A Y= 2c m (b) 4c m (c) 6c m (d) 8c m