The length of the latus rectum of $9x^2 + 25y^2 - 90x - 150y + 225 = 0$ is

If $l$ and $b$ are respectively the length and breadth of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$,then $(l, b) =$

If $F_1$ and $F_2$ are the foci of the ellipse $16 x^2+25 y^2=400$ and $P$ is any point on it,then the value of the product $P F_1 \cdot P F_2$ lies in the interval

If the eccentricities of the two ellipses $\frac{x^2}{169} + \frac{y^2}{25} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are equal,then the value of $a/b$ is

Let the line $2x + 3y - k = 0, k > 0$,intersect the $x$-axis and $y$-axis at the points $A$ and $B$,respectively. If the equation of the circle having the line segment $AB$ as a diameter is $x^2 + y^2 - 3x - 2y = 0$ and the length of the latus rectum of the ellipse $x^2 + 9y^2 = k^2$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $2m + n$ is equal to

The locus of the midpoints of the intercepted portion of the tangents by the coordinate axes,which are drawn to the ellipse $x^2+2y^2=2$,is