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 $\cos 2 \theta=$

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16x$. If one of these tangents touches the two curves at $Q$ and $R$,then $(QR)^2$ is equal to

Let $a$ and $b$ be non-zero real numbers. Then,the equation $(a x^2+b y^2+c)(x^2-5 x y+6 y^2)=0$ represents

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 . . . . . . .

The tangent at $A(-1, 2)$ on the circle $x^2+y^2-4x-8y+7=0$ touches the circle $x^2+y^2+4x+6y=0$ at $B$. Then,a point of trisection of $AB$ is

From a point $P(0, b)$,two tangents are drawn to the circle $x^2+y^2=16$. These two tangents intersect the $X$-axis at two points $A$ and $B$. If the area of $\triangle PAB$ is minimum,then the equation of its circumcircle is