[" Let "P(x_(1),y_(1))" and "Q(x_(2),y_(2)),y_(1)<0,y_(2)<0," be the "],[" end points of the latus rectum of the ellipse "],[x^(2)+4y^(2)=4" .The equations of parabolas with "],[" latus rectum "PQ" are : "],[" (1) "x^(2)+2sqrt(3)y=3+sqrt(3)],[" (2) "x^(2)-2sqrt(3)y=3+sqrt(3)],[" (3) "x^(2)+2sqrt(3)y=3-sqrt(3)],[" (4) "x^(2)-2sqrt(3)y=3-sqrt(3)]

Property 4: The equation of the family of circles passing through two given points P(x_(1),y_(1)) and Q(x_(2),y_(2))

Find the distance between the points P(x_(1),y_(1),z_(1)) and Q(x_(2),y_(2),z_(2))

Distance between P (x_(1), y_(1)) and Q (x_(2), y_(2)) when PQ is parallel to y - axis is

The line y=mx+1 touches the curves y=-x^(4)+2x^(2)+x at two points P(x_(1),y_(1)) and Q(x_(2),y_(2)) . The value of x_(1)^(2)+x_(2)^(2)+y_(1)^(2)+y_(2)^(2) is

The line y=mx+1 touches the curves y=0x^(4)+2x^(2)+x at two points P(x_(1),y_(1)) and Q(x_(2),y_(2)) . The value of x_(1)^(2)+x_(2)^(2)+y_(1)^(2)+y_(2)^(2) is

The line y=mx+1 touches the curves y=-x^(4)+2x^(2)+x at two points P(x_(1),y_(1)) and Q(x_(2),y_(2)) . The value of x_(1)^(2)+x_(2)^(2)+y_(1)^(2)+y_(2)^(2) is

For points P -= (x_(1) ,y_(1)) and Q = (x_(2),y_(2)) of the coordinate plane , a new distance d (P,Q) is defined by d(P,Q) = |x_(1)-x_(2)|+|y_(1)-y_(2)| Let O -= (0,0) ,A -= (1,2), B -= (2,3) and C-= (4,3) are four fixed points on x-y plane Let R(x,y) such that R is equidistant from the point O and A with respect to new distance and if 0 le x lt 1 and 0 le y lt 2 , then R lie on a line segment whose equation is