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

At the point P(a , a^n) on the graph of y=x^n ,(n in N), in the first quadrant, a normal is drawn. The normal intersects the y-a xi s at the point (0, b)dot If ("lim")_(avec0)b=1/2, then n equals 1 (b) 3 (c) 2 (d) 4

The tangent drawn at the point (0, 1) on the curve y=e^(2x) meets the x-axis at the point -

Determine p such that the length of the sub-tangent and sub-normal is equal for the curve y=e^(p x)+p x at the point (0,1)dot

If three normals are drawn from the point (c, 0) to the parabola y^(2)= x , then-

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

From a point (d, 0) three normals are drawn to the parabola y^(2)=x , 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

Tangent drawn from the point P(4,0) to the circle x^2+y^2=8 touches it at the point A in the first quadrant. Find the coordinates of another point B on the circle such that A B=4 .

If tangents are drawn to y^2=4a x from any point P on the parabola y^2=a(x+b), then show that the normals drawn at their point for contact meet on a fixed line.

Tangent and normal are drawn at the point P-=(16 ,16) of the parabola y^2=16 x which cut the axis of the parabola at the points A and B , respectively. If the center of the circle through P ,A and B is C , then the angle between P C and the axis of x is