if `a, b, c, a_1, b_1 and c_1` are non-zero complexnumbers satisfying `a/a_1+b/b_1+c/c_1=1+i and a_1/a+b_1/b+c_1/c=0,` where `i=sqrt-1,` the value of `a^2/a_1^2+b^2/b_1^2+c^2/c_1^2` is

Show that the parallelogram formed by ax+by+c=0, a_1 x+ b_1 y+c=0, ax+by+c_1 =0 and a_1 x+b_1 y+c_1 =0 will be a rhombus if a^2 +b^2 = (a_1)^2 + (b_1)^2 .

For two linear equations , a_1x + b_1y + c_1=0 and a_2x + b_2y + c_2 = 0 , the condition (a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2) is for.

If the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

If ax^2+2bx+c=0 and a_1x^2+2b_1x+c_1=0 have commonroot and a/a_1,b/b_1,c/c_1 are in A.P., show that are: ax^2+2bx+c=0

Find the condition so that the two equations a_1 x^2+b_1 x+c_1=0 and a_2 x^2+b_2 x+c_2=0 will have a common root.

If the points (a_1, b_1),\ \ (a_2, b_2) and (a_1+a_2,\ b_1+b_2) are collinear, show that a_1b_2=a_2b_1 .

The condition for the lines with direction ratio a_1,b_1,c_1 and a_2,b_2,c_2 are perpendicular is -

If a^2+2bx+c=0 and a_1x^2+2b_1x+c_1=0 have a common root and a/a_1,b/b_1,c/c_1 are in AP then a_1,b_1,c_1 are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

If A_1,B_1,C_1, , are respectively, the cofactors of the elements a_1, b_1c_1, , of the determinant "Delta"=|a_1b_1c_1a_1b_2c_2a_3b_3c_3|,"Delta"!=0 , then the value of |B_2C_2B_3C_3| is equal to a1^2Δ b. a_1Δ c. a_1^2Δ d. a1 2^2Δ

If A_1,B_1,C_1, , are respectively, the cofactors of the elements a_1, b_1c_1, , of the determinant "Delta"=|a_1b_1c_1a_1b_2c_2a_3b_3c_3|,"Delta"!=0 , then the value of |B_2C_2B_3C_3| is equal to a1 2 b. a_1 c.a_1^2 d. a1 2^2