If `t_(1)` and `t_(2)` are the lengths of tangents drawn on circle with radius `r` from two conjugate points `A,B` then `t_(1)^(2)+t_(2)^(2)=`

From any point P tangents of length t_(1) and t_(2) are drawn to two circles with centre A,B and if PN is the perpendicular from P to the radical axis and t_(1)^(2) - t_(2)^(2) = K.PN*AB then K=

A particle is thrown such that for two angles of projection range R is same. If T_(1) and T_(2) are the time of flights and H_(1) and H_(2) are the maximum hrights corresponding to these angles, find (a) (T_(1)T_(2))/(R ) and (b) (4sqrt(H_(1)H_(2)))/(R ) .

t_(1),t_(2),t_(3) are lengths of tangents drawn from a point (h,k) to the circles x^(2)+y^(2)=4,x^(2)+y^(2)-4=0andx^(2)+y^(2)-4y=0 respectively further, t_(1)^(4)=t_(2)^(2)" "t_(3)^(2)+16 . Locus of the point (h,k) consist of a straight line L_(1) and a circle C_(1) passing through origin. A circle C_(2) , which is equal to circle C_(1) is drawn touching the line L_(1) and the circle C_(1) externally. The distance between the centres of C_(1)andC_(2) is

t_(1),t_(2),t_(3) are lengths of tangents drawn from a point (h,k) to the circles x^(2)+y^(2)=4,x^(2)+y^(2)-4=0andx^(2)+y^(2)-4y=0 respectively further, t_(1)^(4)=t_(2)^(2)" "t_(3)^(2)+16 . Locus of the point (h,k) consist of a straight line L_(1) and a circle C_(1) passing through origin. A circle C_(2) , which is equal to circle C_(1) is drawn touching the line L_(1) and the circle C_(1) externally. Equation of L_(1) is

t_(1),t_(2),t_(3) are lengths of tangents drawn from a point (h,k) to the circles x^(2)+y^(2)=4,x^(2)+y^(2)-4=0andx^(2)+y^(2)-4y=0 respectively further, t_(1)^(4)=t_(2)^(2)" "t_(3)^(2)+16 . Locus of the point (h,k) consist of a straight line L_(1) and a circle C_(1) passing through origin. A circle C_(2) , which is equal to circle C_(1) is drawn touching the line L_(1) and the circle C_(1) externally. Equation of C_(1) is

The base of a triangle is divided into three equal parts.If t_(1),t_(2),t_(3) are the tangents of the angles subtended by these parts at the opposite vertex,prove that ((1)/(t_(1))+(1)/(t_(2)))((1)/(t_(2))+(1)/(t_(3)))=4(1+(1)/(t_(2)^(2)))

Two tangents TP and TQ are drawn to a circle with centre O from an external point T . Prove that /_PTQ=2/_OPQ .