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 . . . . . . .

Two chords of the circle $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$ pass through the point $(g, h+c)$,and the line $y=x$ bisects these two chords. Then:

If the length of the tangent from any point on the circle $(x-3)^2+(y+2)^2=5r^2$ to the circle $(x-3)^2+(y+2)^2=r^2$ is $16$ units, then the area between the two circles in sq. units is (in $\pi$)

The equation of a common tangent to the circle $x^2+y^2=4$ and the ellipse $2x^2+25y^2=50$ is

Let the line $L: \sqrt{2}x + y = \alpha$ pass through the point of intersection $P$ (in the first quadrant) of the circle $x^2 + y^2 = 3$ and the parabola $x^2 = 2y$. Let the line $L$ touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y$-axis,then the square of the area of the triangle $PQ_1Q_2$ is equal to:

The circle ${x^2} + {y^2} = 4$ cuts the line joining the points $A(1, 0)$ and $B(3, 4)$ in two points $P$ and $Q$. Let $\frac{BP}{PA} = \alpha$ and $\frac{BQ}{QA} = \beta$. Then $\alpha$ and $\beta$ are roots of the quadratic equation