In the List-I each item contains equations of two circles, List-II contains the number of common tangents for each pair of circles given in List-I. Match the items of List-I with those of the items of List-II
| $$ \text { List-I } $$ |
$$ \text { List-II } $$ |
||
|---|---|---|---|
| A. | $$ \begin{aligned} & x^2+y^2+2 x+8 y-23=0 \\ & x^2+y^2-4 x-10 y+19=0 \end{aligned} $$ |
I. | 0 |
| B. | $$ \begin{aligned} & x^2+y^2=1 \\ & x^2+y^2-2 x-6 y+6=0 \end{aligned} $$ |
II. | 1 |
| C. | $$ \begin{aligned} & x^2+y^2-8 x+2 y=0 \\ & x^2+y^2-2 x-16 y+25=0 \end{aligned} $$ |
III. | 2 |
| D. | $$ \begin{aligned} & x^2+y^2=4 \\ & x^2+y^2-2 x=0 \end{aligned} $$ |
IV. | 3 |
| V. | 4 | ||
$$ \text { The correct match is } $$
$\left(0, \frac{3}{4}\right)$ is the radical centre of the circles $S_1: x^2+y^2-2 x+6 y=0, S_2: x^2+y^2+2 g x-2 y+6=0$ and $S_3: x^2+y^2-12 x+2 f y+3=0$. If $S_2$ and $S_3$ intersect orthogonally, then $(g, f)=$
For the circles $(x-a)^2+y^2=a^2$ and $x^2+(y-a)^2=a^2$, where $a>0$, which one of the following is not true?
If $S(a, b)$ is a fixed point and $P(\alpha, \beta)$ is such a variable point that $4\left[(x-a)^2+(y-b)^2\right]=(\alpha x+\beta y+7)^2$ represents a parabola, then the locus of $P(\alpha, \beta)$ is
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