If `x_1, x_2 , x_3` and `y_1 , y_2 , y_3` are both in G.P. with the same common ratio, then the points `(x_1 , y_1),(x_2 , y_2)` and `(x_3 , y_3)`

If x_(1),x_(2),x_(3) as well as y_(1),y_(2),y_(3) are in G.P. with the same common ratio , then show that the points (x_(1),y_(1)),(x_(2),y_(2))and(x_(3),y_(3)) lie on a straight line .

If x_(1),x_(2),x_(3) as well as y_(1),y_(2),y_(3) are in A.P. with the same common difference , then show that the points (x_(1),y_(1)),(x_(2),y_(2))and(x_(3),y_(3)) are collinear.

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

If x ,2y ,3z are in A.P., where the distinct numbers x ,y ,z are in G.P, then the common ratio of the G.P. is a. 3 b. 1/3 c. 2 d. 1/2

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

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 unit , then prove that, |{:(x_1,y_1,2),(x_2,y_2,2), (x_3,y_3,2):}|^2=3a^4

If each of the points (x_1, 4), (-2,y_1) lies on the line joining the points (2, -1), (5, -3), then the points P(x_1, y_1) lies on the line :

Prove that the area of the triangle with vertices (p,q),(x_(1),y_(1)) and (x_(2),y_(2)) where p,x_(1),x_(2) are in G.P. with common ratio r_(1) and q, y_(1), y_(2) and in G.P with common ratio r_(2) is (1)/(2) pq(r_(1)-1) (r_(2)-1) (r_(2)-r_(1)) sq unit.

If the normals to the ellipse x^2/a^2+y^2/b^2= 1 at the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) are concurrent, prove that |(x_1,y_1,x_1y_1),(x_2,y_2,x_2y_2),(x_3,y_3,x_3y_3)|=0 .

If a x_1^2+b y_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_2 x_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)