The roots of the equation `t^3+3a t^2+3b t+c=0a r ez_1, z_2, z_3` which represent the vertices of an equilateral triangle. Then `a^2=3b` b. `b^2=a` c. `a^2=b` d. `b^2=3a`

Let lambda in R . If the origin and the non-real roots of 2z^2+2z+lambda=0 form the three vertices of an equilateral triangle in the Argand plane, then lambda is (a.) 1 (b) 2/3 (c.) 2 (d.) -1

z_1a n dz_2 are the roots of 3z^2+3z+b=0. if O(0),(z_1),(z_2) form an equilateral triangle, then find the value of bdot

If the equation whose roots are the squares of the roots of the cubic x^3-a x^2+b x-1=0 is identical with the given cubic equation, then (a) a=0,b=3 b. a=b=0 c. a=b=3 d. a ,b , are roots of x^2+x+2=0

If the roots of the equation (c^(2)-ab)x^(3)-2(a^(2)-bc)x+b^(2)-ac=0 are real and equal prove that either a=0 (or) a^(3)+b^(3)+c^(3)=3"abc" .

Let alpha,beta be the roots of the quadratic equation a x^2+b x+c=0and Delta =b^2-4ac cdotIfalpha+beta,alpha^2+beta^2,alpha^3+beta^3 are in G.P. Then a. Delta!=0 b. bDelta=0 c. cDelta =0 d. Delta =0

If z_1a n dz_2 are the complex roots of the equation (x-3)^3+1=0,t h e nz_1+z_2 equal to a. 1 b. 3 c. 5 d. 7

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))) .

If the area of the triangle formed by the points (2a ,b)(a+b ,2b+a), and (2b ,2a) is 2qdotu n i t s , then the area of the triangle whose vertices are (a+b ,a-b),(3b-a ,b+3a), and (3a-b ,3b-a) will be_____

If xsina+ysin2a+zsin3a=sin4a xsinb+ysin2b+zsin3b=sin4b , xsinc+ysin2c+zsin3c=sin4c , then the roots of the equation t^3-(z/2)t^2-((y+2)/4)t+((z-x)/8)=0,a , b , c ,!=npi, are (a) sina ,sinb ,sinc (b) cosa ,cosb ,cosc (b) sin2a ,sin2b ,sin2c (d) cos2a ,cos2bcos2c

The vertices of a triangle are [a t_1t_2,a(t_1 +t_2)] , [a t_2t_3,a(t_2 +t_3)] , [a t_3t_1,a(t_3 +t_1)] Then the orthocenter of the triangle is (a) (-a, a(t_1+t_2+t_3)-at_1t_2t_3) (b) (-a, a(t_1+t_2+t_3)+at_1t_2t_3) (c) (a, a(t_1+t_2+t_3)+at_1t_2t_3) (d) (a, a(t_1+t_2+t_3)-at_1t_2t_3)