Three points `A(x_1 , y_1), B (x_2, y_2) and C(x, y)` are collinear. Prove that: `(x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1)`.

A (3,4 ), B (-3, 0) and C (7, -4) are the vertices of a triangle. Show that the line joining the mid-points D (x_1, y_1), E (x_2, y_2) and F (x, y) are collinear. Prove that (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1)

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 points P(h , k),\ Q(x_1, y_1)a n d\ R(x_2, y_2) lie on a line. Show that: (h-x_1)(y_2-y_1)=(k-y_1)(x_2-x_1)dot

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 three points (x_1,y_1),(x_2, y_2),(x_3, y_3) lie on the same line, prove 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 points (x_1,y_1),(x_2,y_2)and(x_3,y_3) are collinear, then the rank of the matrix {:[(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)]:} will always be less than

STATEMENT-1: If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) are collinear, then |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 STATEMENT-2: If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0 then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) will be collinear. STATEMENT-3: If lines a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0 are concurrent then |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0

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

If theta\ is the angle which the straight line joining the points (x_1, y_1)a n d\ (x_2, y_2) subtends at the origin, prove that tantheta=(x_2y_1-x_1y_2)/(x_1x_2+y_1y_2)\ a n dcostheta=(x_1x_2+y_1y_2)/(sqrt(x1 2 x2 2+y2 2x1 2+y1 2x2 2+ y1 2y2 2))

Prove that the line passing through the points (x_(1),y_(1)) and (x_(2),y_(2)) is at a distance of |(x_(1)y_(2)-x_(2)y_(1))/(sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)))| from origin.