14
Question
The equation of the common tangent to the curves is y2=8x and xy=−1 is
The equation of the common tangent to the curves is y2=8x and xy=−1 is
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Solution
The correct option is D
Any point on y2=8x is (2t2,4t) where the tangent is yt=x+2t2.
Solving it with xy=−1,y(yt−2t2)=−1 or ty2−2t2+1=0.
For common tangent, it should have equal roots.
∴4t4−4t=0 ⇒t=0,1.
∴ The common tangent is y=x+2, (when t=0, it is x=0 which can touch xy=-1 at infinity only).
Any point on y2=8x is (2t2,4t) where the tangent is yt=x+2t2.
Solving it with xy=−1,y(yt−2t2)=−1 or ty2−2t2+1=0.
For common tangent, it should have equal roots.
∴4t4−4t=0 ⇒t=0,1.
∴ The common tangent is y=x+2, (when t=0, it is x=0 which can touch xy=-1 at infinity only).