Find the equations of the tangent and the normal to `16 x^2+9y^2=144` at `(x_1,\ y_1)` where `x_1=2` and `y_1>0` .

Options

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`

Answer

A, B, D

Explanation

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)`