`|(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

The value of ,2x_(1)y_(1),x_(1)y_(2)+x_(2)y_(1),x_(1)y_(3)+x_(3)y_(1)x_(1)y_(2)+x_(2)y_(1),2x_(2)y_(2),x_(2)y_(3)+x_(3)y_(2)x_(1)y_(3)+x_(3)y_(1),x_(2)y_(3)+x_(3)y_(2),2x_(3)y_(3)]| is

If x_1 , x_2, x_3 as well as y_1, y_2, y_3 are in A.P., then the points (x_1, y_1), (x_2, y_2), (x_3, y_3) are (A) concyclic (B) collinear (C) three vertices of a parallelogram (D) none of these

If the circle x^2 + y^2 = a^2 intersects the hyperbola xy=c^2 in four points P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4) , then : (A) x_1 + x_2 + x_3 + x_4 = 0 (B) y_1 + y_2 + y_3 + y_4 = 0 (C) x_1 x_2 x_3 x_4= c^4 (D) y_1 y_2 y_3 y_4 = c^4

The value of |(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) +b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))| , is

The differential equation of all conics whose centre klies at origin, is given by (a) (3xy_(2)+x^(2)y_(3))(y-xy_(1))=3xy_(2)(y-xy_(1)-x^(2)y_(2)) (b) (3xy_(1)+x^(2)y_(2))(y_(1)-xy_(3))=3xy_(1)(y-xy_(2)-x^(2)y_(3)) ( c ) (3xy_(2)+x^(2)y_(3))(y_(1)-xy)=3xy_(1)(y-xy_(1)-x^(2)y_(2)) (d) None of these

(x) / (2) + (y) / (3) = 1 (x) / (3) + (y) / (2) = 1

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 the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) be collinear, show that: (y_2 - y_3)/(x_2 x_3) + (y_3 - y_1)/(x_3 x_2) + (y_1 - y_2)/(x_1 x_2) = 0

If |x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1| then the two triangles with vertices (x_1, y_1),(x_2,y_2),(x_3,y_3) and (a_1,b_1),(a_2,b_2),(a_3,b_3) are equal to area (b) similar congruent (d) none of these

A triangle has vertices A_(i) (x_(i),y_(i)) for i= 1,2,3,. If the orthocenter of triangle is (0,0) then prove that |{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|=0