If `A(x_1,y_1),B(x_2,y_2) and C(x_3,y_3)` are the vertices of traingle ABC and `x_(1)^(2)+y_(1)^(2)=x_(2)^(2)+y_(2)^(2)=x_3^(2)+y_(3)^(2)`, then show that `x_1 sin2A+x_2sin2B+x_3sin2C=y_1sin2A+y_2sin2B+y_3sin 2C=0`.

Let C_(1):y=x^(2)sin3x,C_(2):y=x^(2)and C_(3):y=-y^(2), then

If A(x_1, y_1),B(x_2, y_2) and C(x_3,y_3) are vertices of an equilateral triangle whose each side is equal to a , then prove that |[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=3a^4

If the points (x_1, y_1),(x_2,y_2), and (x_3, y_3) are collinear show that (y_2-y_3)/(x_2x_3)+(y_3-y_1)/(x_3x_1)+(y_1-y_2)/(x_1x_2)=0

If A (x_(1), y_(1)), B (x_(2), y_(2)) and C (x_(3), y_(3)) are vertices of an equilateral triangle whose each side is equal to a, then prove that |(x_(1),y_(1),2),(x_(2),y_(2),2),(x_(3),y_(3),2)| is equal to

If A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) are the vertices of a DeltaABC and (x, y) be a point on the median through A . Show that : |(x, y, 1), (x_1, y_1, 1), (x_2, y_2, 1)| + |(x, y, 1), (x_1, y_1, 1), (x_3, y_3, 1)|=0

If x=sin(2tan^(-1)2)and y=sin((1)/(2)tan^(-1).(4)/(3)) , then prove that y^2 = 1 - x

If A(x_1, y_1),B(x_2, y_2),C(x_3, y_3) are the vertices of a triangle, then the equation |x y1x_1y_1 1x_2y_3 1|+|x y1x_1y_1 1x_3y_3 1|=0 represents (a)the median through A (b)the altitude through A (c)the perpendicular bisector of B C (d)the line joining the centroid with a vertex

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0

"If "e^(sin(x^(2)+y^(2)))=tan""(y^(2))/(4)+sin^(-1)x," then y'(0) can be "