Let the points P ,Q lie on ellipse `x^(2)+4y^(2)=4` satisfying the `OP |OQ`,where O is origin the minimum value of `OP.OQ` is `alpha` find 5`alpha`

If y=x+c touches the ellipse 3x^(2)+4y^(2)=12 at the point P, then the value of the length OP (where O is the origin) is equal to

P_(1) and P_(2) are corresponding points on the ellipse (x^(2))/(16)+(y^(2))/(9)=1 and its auxiliary circle respectively. If the normal at P_(1) to the ellipse meets OP_(2) in Q (where O is the origin), then the length of OQ is equal to

Two points P and Q in a plane are related if OP=OQ, where is a fixed point.This relation is :

Find the point P on the circle x^2+y^2 -4x-6y+9=0 such that (i) /_POX is minimum (ii) OP is maximum, where O is the origin and OX is the x-axis.

A straight line l with negative slope passes through (8,2) and cuts the coordinate axes at P and Q. Find absolute minimum value of OP+ OQ where O is the origin-

A straight line L with negative slope passes through the point (8,2) and cuts the positive coordinate axes at the points P and Q .as L varies, the absolute minimum value of (OP+OQ)/2 O is origin is

If O is the origin and OP, OQ are distinct tangents to the circle x^(2)+y^(2)+2gx+2fy+c=0 then the circumcentre of the triangle OPQ is

if P,Q are two points on the line 3x+4y+15=0 such that OP=OQ=9 then the area of triangle OPQ is