If `(x_1-x_2)^2+(y_1-y_2)^2=a^2`, `(x_2-x_3)^2+(y_2-y_3)^2=b^2`, `(x_3-x_1)^2+(y_3-y_1)^2=c^2`, and `k|[x_1,y_1, 1],[x_2,y_2, 1],[x_3,y_3, 1]|=(a+b+c)(b+c-b)(c+a-b)xx(a+b-c)` , then the value of `k` is `1` b. `2` c. `4` d. none of these

If (x_1-x_2)^2+(y_1-y_2)^2=a^2,(x_2-x_3)^2+(y_2-y_3)^2=b^2 and (x_3-x_1)^2+(y_3-y_1)^2=c^2 then prove that 4|{:(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1):}|= (a+b+c)(b+c-a)(c+a-b)(a+b-c)

If (x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2)(x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2)(x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2) and 2s=a+b+c then (1)/(4)det[[x_(2),y_(2),1x_(3),y_(3),1]]^(2)

If (x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)=144,(x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=25 and (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=169, then the value of det[[x_(1),y_(1),1x_(2),y_(2),1x_(3),y_(3),1]]^(2) is 30(b)30^(2)(c)60(d)60^(2)

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)

If x!=y!=za n d|[x,x^2, 1+x^3],[y ,y^2 ,1+y^3],[z, z^2, 1+z^3]|=0, then the value of x y z is a.1 b. 2 c. -1 d. 2

If y=3^(x-1)+3^(-x-1), then the least value of y is (a)2(b)6(c)2/3(d)3/2

|(2x_1y_1, x_1y_2+x_2y_1, x_1y_3+x_3y_1),(x_1y_2+x_2y_1, 2x_2y_2, x_2y_3+x_3y_2),(x_1y_3+x_3y_1, x_2y_3+x_3y_2, 2x_3y_3)|= (A) 0 (B) 1 (C) -1 (D) none of these

A triangle has its three sides equal to a , b and c . If the coordinates of its vertices are A(x_1, y_1),B(x_2,y_2)a n dC(x_3,y_3), show that |[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=(a+b+c)(b+c-a)(c+a-b)(a+b-c)