If $A = \{(x, y) : x^2 + y^2 = 25\}$ and $B = \{(x, y) : x^2 + 9y^2 = 144\}$,then $A \cap B$ contains

Match the conics in Column $I$ with the statements/expressions in Column $II$.

Column $I$	Column $II$
$(A)$ Circle	$(p)$ The locus of the point $(h, k)$ for which the line $h x+k y=1$ touches the circle $x^2+y^2=4$
$(B)$ Parabola	$(q)$ Points $z$ in the complex plane satisfying $|z+2|-|z-2|= \pm 3$
$(C)$ Ellipse	$(r)$ Points of the conic have parametric representation $x=\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right), y=\frac{2 t}{1+t^2}$
$(D)$ Hyperbola	$(s)$ The eccentricity of the conic lies in the interval $1 \leq x < \infty$
	$(t)$ Points $z$ in the complex plane satisfying $\operatorname{Re}(z+1)^2=|z|^2+1$

$A$ bar of length $20$ units moves with its ends on two fixed straight lines at right angles. $A$ point $P$ marked on the bar at a distance of $8$ units from one end describes a conic whose eccentricity is

If two tangents drawn from a point $(\alpha, \beta)$ lying on the ellipse $25x^{2} + 4y^{2} = 1$ to the parabola $y^{2} = 4x$ are such that the slope of one tangent is four times the other,then the value of $(10\alpha + 5)^{2} + (16\beta^{2} + 50)^{2}$ equals

Suppose $a, b$ are real numbers such that $ab \neq 0$. Which of the following four figures represents the curve $(y-ax-b)(bx^2+ay^2-ab)=0$?

The equation of a common tangent to the curves $y^2 = 16x$ and $xy = -4$ is: