$A$ quadratic polynomial $y = f(x)$ with constant term $3$ neither touches nor intersects the $x$-axis and is symmetric about the line $x = 1$. The coefficient of the leading term of the polynomial is unity. $A$ point $A(x_1, y_1)$ with abscissa $x_1 = 1$ and a point $B(x_2, y_2)$ with ordinate $y_2 = 11$ are given in a Cartesian rectangular system of coordinates $OXY$ in the first quadrant on the curve $y = f(x)$,where $O$ is the origin. The vertex of the quadratic polynomial is:
- A$(1, 1)$
- B$(2, 3)$
- C$(1, 2)$
- DNone of these
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| List-$I$ | List-$II$ |
|---|---|
| $(A)$ $\left[\frac{p}{2}\left(t+\frac{1}{t}\right), \frac{q}{2}\left(t-\frac{1}{t}\right)\right]$ | $(I)$ parabola |
| $(B)$ $(p+q \cos \theta, r+q \sin \theta)$ | $(II)$ circle |
| $(C)$ $(p+\lambda^2, q-\lambda)$ | $(III)$ ellipse |
| $(IV)$ hyperbola |
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