If the point `P[X_1+t(X_2-X_1),y_1+t(y_2-y_1]` divides AB internally where `(X_1,Y_1)` and B B(`X_2,Y_2`) then, if `tin (0,k)` , find k

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 :

If the position coordinates of the points A and B are (x_1,y_1,z_1) and (x_2,y_2,z_2) respectively , determine the magnitude and direction of the vector vec(AB) .

If y=e^x , Prove that , xy_2-(2x-1)y_1+(x-1)y=0 .

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x_1,y_1) and (x_2,y_2), respectively, then (a) x_1=y_2 (b) x_1=y_1 (c) y_1=y_2 (d) x_2=y_1

The coordinates of the ends of a focal chord of the parabola y^2=4a x are (x_1, y_1) and (x_2, y_2) . Then find the value of x_1x_2+y_1y_2 .

If the join of (x_1,y_1) and (x_2,y_2) makes on obtuse angle at (x_3,y_3), then prove that (x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)<0

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if 0 lt k lt 1

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if k=0

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if k=(1)/(2) ?

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if k gt 1