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+y1 2x2 2+y2 2))`

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))

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

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

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 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 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

If the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) be collinear, show that: (y_2 - y_3)/(x_2 x_3) + (y_3 - y_1)/(x_3 x_2) + (y_1 - y_2)/(x_1 x_2) = 0

If the image of the co-ordinate of a point (x_1,y_1) is (x_2,y_2) with respect to the straight line lx+my+n=0,then show that (x_2-x_1)/l=(y_2-y_1)/m=(-2(lx_1+my_1+n))/(l^2+m^2)

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

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