The ratio in which the line segment joining `P(x_1,\ y_1)` and `Q(x_2,\ y_2)` is divided by x-axis is (a) `y_1: y_2` (b) ` y_1: y_2` (c) `x_1: x_2` (d) ` x_1: x_2`

If the mid-point of the segment joining A(x ,\ y+1) and B(x+1,\ y+2) is C(3/2,5/2) , find x ,\ y .

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

If the line segment joining the points P(x_1, y_1)a n d\ Q(x_2, y_2) subtends an angle alpha at the origin O, prove that : O PdotO Q\ cosalpha=x_1x_2+y_1y_2dot

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)

The equation to the chord joining two points (x_1,y_1) and (x_2,y_2) on the rectangular hyperbola xy=c^2 is: (A) x/(x_1+x_2)+y/(y_1+y_2)=1 (B) x/(x_1-x_2)+y/(y_1-y_2)=1 (C) x/(y_1+y_2)+y/(x_1+x_2)=1 (D) x/(y_1-y_2)+y/(x_1-x_2)=1

If the line joining the points (_x_1, y_1) and (x_2, y_2) subtends a right angle at the point (1,1), then x_1 + x_2 + y_2 + y_2 is equal to

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))

Show that the plane ax+by+cz+d=0 divides the line joining (x_1, y_1, z_1) and (x_2, y_2, z_2) in the ratio of (-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))

Statement 1:If the point (2a-5,a^2) is on the same side of the line x+y-3=0 as that of the origin, then a in (2,4) Statement 2: The points (x_1, y_1)a n d(x_2, y_2) lie on the same or opposite sides of the line a x+b y+c=0, as a x_1+b y_1+c and a x_2+b y_2+c have the same or opposite signs.

Show that the plane a x+b y+c z+d=0 divides the line joining the points (x_1, y_1, z_1)a n d(x_2, y_2, z_2) in the ratio (a x_1+b y_1+c z_1+d)/(a x_2+b y_2+c z_2+d) .