Two parabolas with a common 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 the common tangent to the two parabolas is?

Step-by-step solution

As origin is the only common point to x- axis and y -axis, so, origin is the common vertex Let the equation of two of parabolas be y^2=4 a x and x^2=4 b y Now latus rectum of both parabolas =3 aligned & 4 a=4 b=3 & a=b= 3 4 aligned Two parabolas are y^2=3 x and x^2=3 y Suppose y=m x+c is the common tangent. aligned & y^2=3 x (m x+c)^2=3 x & m^2 x^2+(2 m c-3) x+c^2=0 aligned As, the tangent touches at one point only So, b^2-4 a c=0 aligned & (2 m c-3)^2-4 m^2 c^2=0 & 4 m^2 c^2+9-12 m c-4 m^2 c^2=0 aligned aligned &