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_(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 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)

Find the coordinates of the centroid fof the triangle whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)) and (x_(3),y_(3),z_(3))

If (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))and(x_(4),y_(4)) be the consecutive vertices of a parallelogram , show that , x_(1)+x_(3)=x_(2)+x_(4)andy_(1)+y_(3)=y_(2)+y_(4) .

Show that the equation of the chord of the parabola y^(2) = 4ax through the points (x_(1),y_(1)) and (x_(2),y_(2)) on it is (y-y_(1))(y-y_(2)) = y^(2) - 4ax

If y= (tan^(-1)x)^(2) , show that (x^(2)+1)^(2) y_(2)+2x(x^(2)+1)y_(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

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.

The slope of the tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point ( x_(1),y_(1)) is-