17. Equation of common tangent to the curves  y²=8x and xy=-1 is

A) 3y=9x+2

B) y=2x+1

C) 2y=x+8

D) y=x+2

eq of tangent to y²=8x at (a,b) is by=4x+4a and the eq of the tangent to xy=-1 at (c,d)  is dx+cy=-2 now for a common tangent the t equations have to be same so bu comparing the coefficient we get 4/d=-b/c=2a now u have the followind relations b²=8a,cd=-1,ad=2,2ac=-b,4c=-bd now what i would suggest is......not to solve for  a,b,c,d which u can do and which is a proper method....insted u can just guess an option and see if it satisfies the conditions i got (d) as the ans......in fact that was the second option which i tried....

Equation of tangent to y² = 8x at (4t,2t²) is ty = x + 2t²

Now solve this tangent and curve xy = -1 to find the their point of intersection.

(ty - 2t²)y = -1

ty² - 2t²y + 1 = 0

Since this is also a tangent to xy = -1 it would cut the curve at one point and hence the roots of above equation must be equal i.e. discriminant = 0

4t⁴ = 4t

which gives t = 1

hence equation common tangent is y = x + 2

correct option is d.