The line `x = 8` is the directrix of the ellipse `E : (x^2)/(a^2)+(y^2)/(b^2)` with the corresponding focus `(2, 0)`. If the tangent to `E` at the point `P` in the first quadrant passes through the point `(0,4sqrt3)` and intersects the x-axis at Q, then `(3PQ)^2` is equal to

Let A(alpha,0) and B(0,beta) be the points on the line 5x+7y=50 .Let the point divide the line segment "AB" internally in the ratio "7:3" .Let 3x-25=0 be a directrix of the ellipse E:(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and the corresponding focus be "S" .If from "S", the perpendicular on the "x" -axis passes through "P" ,then the length of the latus rectum of "E" is equal to,

The y-axis is the directrix of the ellipse with eccentricity e=(1)/(2) and the corresponding focus is at (3,0), equation to its auxilary circle is

The line 2x+y=3 intersects the ellipse 4x^(2)+y^(2)=5 at two points. The point of intersection of the tangents to the ellipse at these point is

The equation of the line passing through the point (2, 3) and the point of intersection of lines 2x-3y+7=0 and 7x+4y+2=0 is

The line passing through the point of intersection of x+y=2,x-y=0 and is parallel to x+2y=5, is

A line passes through the point of intersection of 2x+y=5 and x+3y+8=0 and paralled to the 3x+4y=7 is

The equation of the line passing through (0,0) and intersection point of 3x-4y=2 and x+2y=-4 is