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 y2=4ax and x2=4by Now latus rectum of both parabolas =3∴4a=4b=3⇒a=b=
3
4
∴ Two parabolas are y2=3x and x2=3y Suppose y=mx+c is the common tangent. ∴y2=3x⇒(mx+c)2=3x⇒m2x2+(2mc−3)x+c2=0 As, the tangent touches at one point only So,b2−4ac=0 ⇒(2mc−3)2−4m2c2=0 ⇒4m2c2+9−12mc−4m2c2=0 ⇒c=
9
12m
=
3
4m
. . . (i) ∴x2=3y⇒x2=3(mx+c)⇒x2−3mx−3c=0 Again, b2−4ac=0 ⇒9m2−4(1)(−3c)=0 ⇒9m2=−12c . . . (ii) Form (i) and (ii) m2=