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Consider the hyperbola H:x^2-y^2=1 and a...

Consider the hyperbola `H:x^2-y^2=1` and a circle S with centre `N(x_2,0)` Suppose that H and S touch each other at a point `(P(x_1,y_1)` with `x_1 > 1 and y_1 > 0` The common tangent to H and S at P intersects the x-axis at point M. If (l,m) is the centroid of the triangle `DeltaPMN` then the correct expression is (A) `(dl)/(dx_1)=1-1/(3x_1^2)` for `x_1 > 1` (B) `(dm)/(dx_1) =x_!/(3(sqrtx_1^2-1))) for x_1 > 1` (C) `(dl)/(dx_1)=1+1/(3x_1^2) for x_1 > 1` (D) `(dm)/(dy_1)=1/3 for y_1 > 0`

A

`(dl)/(dx_(1))=1-(1)/(3x_(1)^(2))" for "x_(1) gt 1`

B

`(dm)/(dx_(1))=(x)/(3(sqrt(x_(1)^(2)-1)))" for "x_(1) gt 1`

C

`(dl)/(dx_(1))=1+(1)/(3x_(1)^(2))" for "x_(1) gt 1`

D

`(dm)/(dy_(1))=(1)/(3)" for "y_(1) gt 0`

Text Solution

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The correct Answer is:
A,B,D
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