Three points P (h, k), `Q(x_1, y_1)` and 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)`.

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 (x_1,y_1), (x_2,y_2) and (x_1 + x_2, y_1+y_2) are collinear, prove that x_1y_2 = x_2y_1

A line passes through (x_1, y_1) and (h,k). If slope of the line is m, show that k-y_1=m(h-x_1) .

The line (x-x_1)/0=(y-y_1)/1=(z-z_1)/2 is

A line passes through (x_1, y_1) and (h, k). If slope of line is m, show that k-y_1 = m (h -x_1 ) .

Find the ratio in which the plane ax+by+cz+d=0 divides the join of the points P(x_1,y_1,z_1) and Q(x_2,y_2,z_2) lying on the plane. Hence, show that the points P(1,-2,3) and Q(0,0,-1) lie on opposite sides of the plane 2x+5y+7z=3.

For points P-=(x_1, y_1) and Q-=(x_2, y_2) of the coordinate plane, a new distance d(P ,Q)=|x_1-x_1|+|y_1-y_2| . Let O=(0,0) and A=(3,2) . Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from O and A consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

If y = sin^-1x , prove that (1-x^2)y_2 - x y_1= 0

Prove that the st. line ax + by + c = 0 divides the join of (x_1, y_1) and (x_2, y_2) in the ratio -(ax_1 + by_1 +c)/(ax_2+ by_2+c) .

The equation to the chord joining two points (x_1,y_1)a n d(x_2,y_2) on the rectangular hyperbola x y=c^2 is: x/(x_1+x_2)+y/(y_1+y_2)=1 x/(x_1-x_2)+y/(y_1-y_2)=1 x/(y_1+y_2)+y/(x_1+x_2)=1 (d) x/(y_1-y_2)+y/(x_1-x_2)=1