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

Origin(0,0)is the only point common to x-axis and y-axis.⇒Origin(0,0)is the common vertexLet the equation of2parabola bey2=4axandx2=4byLatus rectum=3⇒4a=4b=3⇒a=b=34∴The2parabolas arey2=3xandx2=3yLety=mx+cbe the common tangenty2=3x⇒(mx+c)2=3x⇒m2x2+(2mc𕒷)x+c2=0The tangent touches at only one point⇒b2𕒸ac=0⇒(2mc𕒷)2𕒸m2c2=0⇒4m2c2+9󔼔mc𕒸m2c2=0⇒c=912m=34m………(1)m2=−c=󔼪mx2=3y⇒x2=3(mx+c)⇒x2𕒷mx𕒷c=0Tangent touches at only one point⇒b2𕒸ac=0⇒9m2𕒸(1)(𕒷c)=0⇒9m2=󔼔c…………(2)From(1)and(2)m2=𕒸c3=󔼳(34m)⇒m3=𕒵⇒m=𕒵⇒c=󔼪∴y=mx+c=−x󔼪⇒4(x+y)+3=0