The co-ordinates of two points P and Q are `(x_1,y_1)` and `(x_2,y_2)` and O is the origin. If circles be described on OP and OQ as diameters then length of their common chord is

The co-ordinates of end points of a focal chord of an ellipse are (x_1,y_1) and (x_2,y_2) .Prove that y_1y_2+4x_1x_2=0

The co-ordinates of the point on the circle x^2+y^2-12x+30=0 which is farthest from the origin are

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:

If O is the origin and if the coordinates of any two points Q_1 \ a n d \ Q_2 are (x_1,y_1) \ a n d \ (x_2,y_2), respectively, prove that O Q_1.O Q_2cos/_Q_1O Q_2=x_1x_2+y_1y_2 .

Through the vertex O of the parabola y^2=4a x , two chords O Pa n dO Q are drawn and the circles on OP and OQ as diameters intersect at Rdot If theta_1,theta_2 , and varphi are the angles made with the axis by the tangents at P and Q on the parabola and by O R , then value of cottheta_1+cottheta_2 is (a) -2tanvarphi (b) -2tan(pi-varphi) (c) 0 (d) 2cotvarphi

Find the co-ordinate of a point on the circle x^2 +y^2 - 4x + 2y - 20 = 0 nearest to (2, 3/2) .

If a chord of a circle x^2 + y^2 =32 makes equal intercepts of length l of the co-ordinates axes, then

P is a point on the line y+2x=1, and Qa n dR two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is

If O be the origin and if coordinates of any two points Q_(1) and Q_(2) be (x_(1),y_(1)) and (x_(2),y_(2)) respectively, prove that overline(OQ_(1))* overline(OQ_(2)) cos angle Q_(1)OQ_(2)=x_(1)x_(2)+y_(1)+y_(2) .

Let A(x_(1),y_(1)) and B(x_(2),y_(2)) be two points on the parabola y^(2) = 4ax . If the circle with chord AB as a dimater touches the parabola, then |y_(1)-y_(2)| is equal to