Two equal parabolas have the same vertex and their axes are at right angles. Prove that their common tangent touches each at the end of their latus recta.

Two equal parabolas have the same vertex and their axes are at right angles.The length of the common tangent to them,is

Two equal parabola have the same vertex and their axes are at right angles. Prove that they cut again at an angle tan^(-1) 3/4 .

Prove that two parabolas having the same focus and their axes in opposite directions, cut at right angles.

In the given figure two circles touch each other at the point C. Prove that the common tangent at P and Q.

Two parabola with a coomon vertex and with axes along x-axis and y-axis, respectively intersect each other in the first quadrant . If the length of the latus rectum of each parabola is 3 , then the equation of common tangent to the two parabola is

Two unequal parabolas have the same common axis which is the x-axis and have the same vertex which is the origin with their concavities in opposite direction.If a variable line parallel to the common axis meet the parabolas in P and P' the locus of the middle point of PP' is

If the bisector of angle A P B , where P Aa n dP B are the tangents to the parabola y^2=4a x , is equally, inclined to the coordinate axes, then the point P lies on the tangent at vertex of the parabola directrix of the parabola circle with center at the origin and radius a the line of the latus rectum.

Two circles of radii a and b touch each other externally and theta be the angle between their direct common tangents (a>b>=2) then