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

Consider the hyperbola `H : x^2-y^2=1` and a circle `S` with center `N(x_2,0)` . Suppose that `Ha n dS` touch each other at a point `P(x_1, y_1)` with `x_1>1` and `y_1> 0.` The common tangent to `Ha n dS` at `P` intersects the x-axis at point `Mdot` If `(l ,m)` is the centroid of the triangle `P M N ,` then the correct expression(s) is (are) `(d l)/(dx_1)=1-1/(3x1 2)forx_1>1` `(d m)/(dx_1)=(x_1)/(3sqrt(x1 2-1))` for`x_1>` `(d l)/(dx_1)=1+1/(3x1 2)` for `x_1>1` `(d m)/(dy_1)=1/3` for `x_1>0`

A

`(dl)/(dx_(1))=1-(1)/(3x_(1)^(2))" for "x_(1)gt1`

B

`(dm)/(dx_(1))=(x_(1))/(3sqrt(x_(1)^(2))-1)" for "x_(1)gt1`

C

`(dl)/(dx_(1))=1+(1)/(3x_(1)^(2))" for "x_(1)gt1`

D

`(dm)/(dy_(1))=(1)/(3)" for "x_(1)gt0`

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
As shown in figure, hyperbola and cirlce touch at `P(x_(1),y_(1))`.
Equation of tangent to H at P is `xx_(1)-yy_(1)=1`.
It meets the x-axis at `M(1//x_(1)=0)`.
Now, centroid of `DeltaPMN` is (l,m).

So," "`l=(x_(1)+x_(2)+(1)/(x_(1)))/(3)and m=(y_(1))/(3)=(sqrt(x_(1)^(2)-1))/(3)`
`"Now, "(dy)/(dx)|._("H at P")=(dy)/(dx)|_("S at P")`
`rArr" "(x_(1))/(y_(1))=(x_(2)-x_(1))/(y_(1))`
`rArr" "x_(2)=2x_(1)`
So, `l=x_(1)+(1)/(3x_(1))`
`(dl)/(dx_(1))=1-(1)/(3x_(1)^(2)),(dm)/(dy_(1))=(1)/(3),(dm)/(dx_(1))=(x_(1))/(3sqrt(x_(1)^(2))-1)`
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