The straight line joining the points `p(x_(1),y_(1))andQ(x_(2),y_(2))` makes an angle `theta` with the positiv direcction of the x-axis prove that , `x_(2)=x_(1)+rcostheta and y_(2)=y_(1)+rsintheta` where `r=overline(PQ)`.

If the line - segment joining the points A(x_(1),y_(1))andB(x_(2),y_(2)) subtends an angle alpha at the origin O, then find the value of cosalpha .

Show that the equation of the straight line joining the points (x_(1),y_(1))and(x_(2),y_(2)) can be expressed in the form (x-x_(2))(y-y_(1))=(x-x_(1))(y-y_(2)) .

If the equation of the pair of straight lines passing through the point (1,1) , one making an angle theta with the positive direction of the x-axis and the other making the same angle with the positive direction of the y-axis, is x^2-(a+2)x y+y^2+a(x+y-1)=0,a!=2, then the value of sin2theta is

The points P divides the line - segment joining the points A(x_(1),y_(1))andB(x_(2),y_(2)) in the ratio m : n . If P lies on the line ax+by +c=0 , prove that, (m)/(n)=-(ax_(1)+by_(1)+c)/(ax_(2)+by_(2)+c) .

Find the distance between P (x_(1), y_(1)) " and " Q (x_(2), y_(2)) when : (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis.

The straight line passing through P(x_1,y_1) and making an angle alpha with x-axis intersects Ax + By + C=0 in Q then PQ =________________

The angle between the line joining the points (1,-2) , (3,2) and the line x+2y -7 =0 is

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

A straight line through the point (3, -2) is inclined at an angle 60^(@) to the line sqrt3x+y=1 . If it intersects the X-axis, then its equation will be