Consider the hyperbola H : x²-y² = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 1 and y1 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 PMN, then the correct expression(s) is(are)

Question

Consider the hyperbola H : x²-y² = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 > 1 and y1 > 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 PMN, then the correct expression(s) is(are)

Vikas TU · Answer

Dear Student,

Equation tangent to H at P is xx1-yy1=1

l=(x1+x2+1/x1)/3 and m=y1/3=sqrt(x1^2-1)/3

now dy/dx(H) at P =dy/dx(S) at P

=>x1/y1 =(x2-x1)/y1

Therefore, l=x1+1/3x1

so, dl/dx1=1- 1/(3x1^2) ,

dm/dx1=1/3

and dm/dx1 =(1/3)*x1/(sqrt(x1^2-3)) [Ans]

Cheers!!

Regards,

Vikas (B. Tech. 4^th year
Thapar University)

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