The locus of the moving point $P$,such that $2PA = 3PB$ where $A$ is $(0,0)$ and $B$ is $(4,-3)$,is

$A$ bottle in the shape of a right-circular cone with height $h$ contains some water. When its base is placed on a flat surface,the height of the vertex from the water level is $a$ units. When it is kept upside down,the height of the base from the water level is $\frac{a}{4}$ units. Then the ratio $\frac{h}{a}$ is

The radius of the circle with minimum area that touches the curve $y = 4 - x^2$ and the lines $y = |x|$ is:

The locus of the midpoint of the chord of contact of tangents drawn from a point on the line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is:

If $A(1, 1)$,$B(-1, 1)$,and $C(-1, -1)$ are three points and a point $P(x, y)$ moves such that $PA^2 = PB^2 + PC^2$,then the equation of the locus of $P$ is:

Find the locus of the midpoint of the chord of the circle $x^2 + y^2 = 16$ which is a tangent to the hyperbola $9x^2 - 16y^2 = 144$.