P is a variable point on the curve y= f(x) and A is a fixed point in the plane not lying on the curve. If `PA^(2)` is minimum, then the angle between PA and the tangent at P is

Tangents are drawn from the point (4, 2) to the curve x^(2)+9y^(2)=9 , the tangent of angle between the tangents :

Let P be a point on the curve c_(1):y=sqrt(2-x^(2)) and Q be a point on the curve c_(2):xy=9, both P and Q be in the first quadrant. If d denotes the minimum distance between P and Q, then d^(2) is ………..

The minimum distance between a point on the curve y = e^x and a point on the curve y=log_e x is

Prove that the curve y=e^(|x|) cannot have a unique tangent line at the point x = 0. Find the angle between the one-sided tangents to the curve at the point x = 0.

A variable circle C touches the line y = 0 and passes through the point (0,1). Let the locus of the centre of C is the curve P. The area enclosed by the curve P and the line x + y = 2 is

The slope of the tangent to the curve y=x^(2) -x at the point where the line y = 2 cuts the curve in the first quadrant, is

The point on the curve f (x) = sqrt(x ^(2) - 4) defined in [2,4] where the tangent is parallel to the chord joining the end points on the curve is

A curve y=f(x) passes through point P(1,1) . The normal to the curve at P is a (y-1)+(x-1)=0 . If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is

If P is any point on the plane l x+m y+n z=pa n dQ is a point on the line O P such that O P.O Q=p^2 , then find the locus of the point Qdot

If A and B are two fixed points and P is a variable point such that PA + PB = 4 , the locus of P is