Consider the hyperbola `H:x^2-y^2=1` and a circile S
with center `N(x_2,0)`. Suppose that H and S touch each
other at a point `P(x_1,y_1)` with `x_1gt1andy_1gt0`.
The common langent to H and S intersects the x-axis at point
M. If (l,m) is the centroid of the triangle `trianglePMN`, 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 ans S touch each other at a point P(x_(1),y_(1)) with x_(1)gt1 and y_(1)gt0 . The common tangent to H ans 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)-

Let the parabolas y=x(c-x)and y=x^(2)+ax+b touch each other at the point (1,0), then-

Let P(6,3) be a point on the hyperbola parabola x^2/a^2-y^2/b^2=1 If the normal at the point intersects the x-axis at (9,0), then the eccentricity of the hyperbola is

Let P(4,3) be a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 . If the normal at P intersects the x-axis at (16,0), then the eccentriclty of the hyperbola is

Consider two curves C1:y^2=4x ; C2=x^2+y^2-6x+1=0 . Then, (a) C1 and C2 touch each other at one point (b) C1 and C2 touch each other exactly at two point (c) C1 and C2 intersect(but do not touch) at exactly two point (d) C1 and C2 neither intersect nor touch each other

Find the coordinates of the centroid of the triangle, formed by the lines x+2y-5=0, y+2x-7=0 and x-y+1=0 .

If the normal at a point P to the hyperbola x^2/a^2 - y^2/b^2 =1 meets the x-axis at G , show that the SG = eSP.S being the focus of the hyperbola.

A triangle has two sides y=m_1x and y=m_2x where m_1 and m_2 are the roots of the equation b alpha^2+2h alpha +a=0 . If (a,b) be the orthocenter of the triangle, then find the equation of the third side in terms of a , b and h .

The line 2x + y = 1 is tangent to the hyperbola x^2/a^2-y^2/b^2=1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

The line 2x + y = 1 is tangent to the hyperbola x^2/a^2-y^2/b^2=1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is