Prove that :
`{:|(a_1x_1+b_1y_1,a_1x_2+b_1y_2,a_1x_3+b_1y_3),(a_2x_1+b_2y_1,a_2x_2+b_2y_2,a_2x_3+b_2y_3),(a_3x_1+b_3y_1,a_3x_2+b_3y_2,a_3x_3+b_3y_3)|`

If a x_1^2+by_1^2+c z_1^2=a x_2^2+b y_2^2+c z_2^2=a x_3 ^2+b y_3 ^2+c z_3^ 2=d a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1= a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that | x_1 y_1 z_1 x_2 y_2 z_2 x_3 y_3 z_3 |= (d-f){(d+2f)/(a b c)}^(1//2)

If sum_(i-1)^4(x1^2+y 1^2)lt=2x_1x_3+2x_2x_4+2y_2y_3+2y_1y_4, the points (x_1, y_1),(x_2,y_2),(x_3, y_3),(x_4,y_4) are

If the circle x^2+y^2=a^2 intersects the hyperbola x y=C^2 at four points P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3), and S(x_4, y_4), then proove x_1+x_2+x_3+x_4=0 , y_1+y_2+y_3+y_4=0 , x_1x_2x_3x_4=C^4 , y_1 y_2 y_3 y_4 = C^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 the vertices of an equilateral triangle are A(x_1,y_1), B(x_2,y_2) and C(x_3,y_3) then show that |(x_1, y_1,2),(x_2, y_2, 2),(x_3,y_3,2)|^2 = 3a^4 , 'a' being the length of each side of the triangle.

A triangle has its three sides equal to a,b and c. If the co-ordinates of its vertices are A(x_1y_1),B(x_2,y_2) and C(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)

Find the coefficient of x in the determinant |{:((1+x)^(a_(1)b_(1)),(1+x)^(a_(1)b_(2)),(1+x)^(a_(1)b_(3))),((1+x)^(a_(2)b_(1)),(1+x)^(a_(2)b_(2)),(1+x)^(a_(2)b_(3))),((1+x)^(a_(3)b_(1)),(1+x)^(a_(3)b_(2)),(1+x)^(a_(3)b_(3))):}|

The three points (x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)), (x_(3), y_(3), z_(3)) are collinear when…………

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.

The differential equation of all conics whose centre k lies 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