If the tangent at the point P(2,4) to the parabola y^2=8x meets the parabola y^2=8x+5 at Qa n dR , then find the midpoint of chord Q Rdot

Let F_(1)(x_(1),0) and F_(2)(x_(2),0) , for x_(1)lt0 and x_(2)gt0 , be the foci of the ellipse (x^(2))/(9)+(y^(2))/(8)=1 . Suppose a parabola having vertex at the orgin and focus at F_(2) intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF_(1)NF_(2) is

The tangent at any point P on the circle x^2+y^2=4 meets the coordinate axes at A and B . Then find the locus of the midpoint of A Bdot

At the point P(a,a^n) on the graph of y=x^n(ninN) in the first quadrant a normal is drawn . The normal Intersects the y-axis at the point (0,b). If lim_(a-0)b=1/2, then n equals

The tangent at any point P onthe parabola y^2=4a x intersects the y-axis at Qdot Then tangent to the circumcircle of triangle P Q S(S is the focus) at Q is

A tangent is drawn to the curve x^(2)(x-y)+a^(2)(x+y)=0 at the origin. Find the angle it makes with the x-axis.

The circle C_(1) : x^(2) + y^(2) = 3 , with centre at O , intersects the parabola x^(2) = 2 y at the point P in the first quadrant . Let the tangent to the circle C-(1) at P touches other two circles C_(2) and C_(3) at R_(2) and R_(3) , respectively. suppose C_(2) and C_(3) have equal radii 2sqrt(3) and centres Q_(2) and Q_(3) , respectively . if Q_(2) and Q_(3) lie on the y-axis , then