Paragraph: Let A be a variable point on locus of feet of perpendicular drawn from focus upon any tangent to the curve |z-2|+|z+2|=6 and B be a variable point on (1-i) z+(1+i) z =10 2 , then Question: If a variable circle touches both the loci on which A and B externally lie then latus rectum of locus of centre of variable circle is:

Step-by-step solution

(I) |z-2|+|z+2|=6 represents an ellipse with major axis 6 and focus (2,0) and (-2,0) A lies on its auxilliary circle i . e., x^2+y^2=9 and (1-i) z+(1+i) z =10 2 (z+ z )-i(z- z )=10 2 2 x-i(2 i y)=10 2 x+y=5 2 B lies on straight line x+y=5 2 A B_ = perpendicular distance O B- radius O A=5-3=2 (II) If a variable circle touches line and circle, then locus will be the Parabola with focus at centre (0,0) of given circle and directrix at a line parallel to given line at a distance of 3 units, equal to radius of circle Di