The equations of the parabolas are y2−4x−8y+40=0⋯(1)
x2−8x−4y+40=0⋯(2)
We observe that if we interchange x and y in equation (1), we obtain equation (2)
So, the two parabolas are symmetric about y=x.
Shortest distance exist between the tangents on both the parabolas which are parallel to y=x
For curve 1, the equation of tangent is
yy1−2(x+x1)−4(y+y1)+40=0
⇒−2x+(y1−4)y−2x1−4y1+40=0
Slope =2y1−4=1
⇒y1=6
Putting value of y1 in eqn (1), we get x1=7
So, (x1,y1)=(7,6)
Since, the parabolas are symmetric about y=x, and if (x1,y1)=(7,6), then (x2,y2)=(6,7)
(∵x and y will interchange)
∴d=√(x1−x2)2+(y1−y2)2=√2