If the line $y=2x+c$ touches the curve $x^2+4y^2=4$,then $c^2=$

Given the base of a triangle and the sum of its other two sides,the locus of the center of its incircle is:

If $4x - 3y + k = 0$ touches the ellipse $5x^{2} + 9y^{2} = 45$,then $k$ is equal to

The tangent to the ellipse $9x^2 + 16y^2 = 288$ making equal intercepts on the co-ordinate axes intersects the $X$-axis and the $Y$-axis in the points $A$ and $B$ respectively. Then $\text{Area}(\triangle OAB) = $ (where $O$ is origin)

The equation $\sqrt{(x-2)^2+y^2}+\sqrt{(x+2)^2+y^2}=4$,where $-2 < x < 2$,represents a

Let $A, B,$ and $C$ be three points on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$. The line joining $A$ and $C$ is parallel to the $x$-axis,and $B$ is the endpoint of the minor axis whose ordinate is positive. Find the maximum area of $\Delta ABC$.