If the coordinates of a variable point `P` are `(acostheta,bsintheta),` where `theta` is a variable quantity, then find the locus of `Pdot`

The coordinates of a moving point P are (a/2(cosectheta+sintheta), b/2(cosectheta-sintheta)), " where " theta is a variable parameter. Show that the equation of the locus P is b^(2)x^(2)-a^(2)y^(2)=a^(2)b^(2) .

The locus of a moving point P(a cos^(3)theta,a sin^(3)theta) is:

The coordinates of a moving point P are ((a)/(2)(co sectheta+sin theta),(b)/(2)(co sec theta-sin theta)) where theta is a variable parameter. Show that the equation of the locus of P is b^(2)x^(2)-a^(2)y^(2)=a^(2)b^(2)

If the intercepts of the variable circle on the x- and yl-axis are 2 units and 4 units, respectively, then find the locus of the center of the variable circle.

If the chord of contact of tangents from a point P to the parabola y^2=4a x touches the parabola x^2=4b y , then find the locus of Pdot

From an external point P , a pair of tangents is drawn to the parabola y^2=4xdot If theta_1a n dtheta_2 are the inclinations of these tangents with the x-axis such that theta_1+theta_2=pi/4 , then find the locus of Pdot

A B is a variable line sliding between the coordinate axes in such a way that A lies on the x-axis and B lies on the y-axis. If P is a variable point on A B such that P A=b ,P b=a , and A B=a+b , find the equation of the locus of Pdot

Let P be a point on the hyperbola x^2-y^2=a^2, where a is a parameter, such that P is nearest to the line y=2xdot Find the locus of Pdot

If R is any point on the x-axis and Q is any point on they y-axis and P is a variable point on RQ with RP=b, PQ=a, then find the equation of locus of P.

An urn contains 5 mangoes and 4 apples. Three fruits are taken at random. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.