Consider the hyperbola `H, x^(2)-y^(2)=1` and a circle S with centre `N(x_, 0)`. Suppose that H and S touch each other at point `P(x_1, y_1)` with `x_1gt1 and y_1gt0`. 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.

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

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

Consider a hyperbola H : x^(2)-y^(2) =k and a parabola P:y=x^(2) then identify the correct statements(S)

Prove that the curves y^2=4x and x^2+y^2-6x+1=0 touch each other at the points (1,\ 2) .

Prove that the curves y^2=4x and x^2+y^2-6x+1=0 touch each other at the points (1,\ 2) .

Find the common tangent of y=1+x^(2) and x^(2)+y-1=0 . Also find their point of contact.

Find the equations of the tangent and the normal to the curve 16x^2+9y^2=145 at the point (x_1,y_1) , where x_1=2 and y_1gt0

Let the line y = mx intersects the curve y^2 = x at P and tangent to y^2 = x at P intersects x-axis at Q. If area ( triangle OPQ) = 4, find m (m gt 0) .

Let the line y = mx intersects the curve y^2 = x at P and tangent to y^2 = x at P intersects x-axis at Q. If area ( triangle OPQ) = 4, find m (m gt 0) .

The tangent to the hyperbola xy=c^2 at the point P intersects the x-axis at T and y- axis at T'.The normal to the hyperbola at P intersects the x-axis at N and the y-axis at N' . The areas of the triangles PNT and PN'T' are Delta and Delta' respectively, then 1/Delta+1/(Delta)' is