The sum of diameters of the circles that touch $(i)$ the parabola $75x^2 = 64(5y - 3)$ at the point $\left(\frac{8}{5}, \frac{6}{5}\right)$ and $(ii)$ the $y$-axis,is equal to $......$

The values of the constant term in the equation of a circle passing through $(1, 2)$ and $(3, 4)$ and touching the line $3x + y - 3 = 0$ are

If $\theta$ is the angle made by the common tangent to the circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ with the positive $X$-axis,then $\cos 2 \theta=$

The focus of the parabola $y^2 = 4x + 16$ is the centre of the circle $C$ of radius $5$. If the values of $\lambda$,for which $C$ passes through the point of intersection of the lines $3x - y = 0$ and $x + \lambda y = 4$,are $\lambda_1$ and $\lambda_2$ (where $\lambda_1 < \lambda_2$),then $12\lambda_1 + 29\lambda_2$ is equal to . . . . . . .

An equilateral triangle $OAB$ is inscribed in the parabola $y^{2}=4x$ with the vertex $O$ at the origin. Then the minimum distance of the circle having $AB$ as a diameter from the origin is:

Let a common tangent to the curves $y^2=4x$ and $(x-4)^2+y^2=16$ touch the curves at the points $P$ and $Q$. Then $(PQ)^2$ is equal to $..........$.