The joint equation of a pair of lines passing through point $(1,4)$, one of which is parallel to X-axis and the other makes an angle of $45^\circ$ with the positive direction of X-axis, is

If a circle passes through the points $(2,3)$ and $(4,5)$ and its center lies on the straight line $y - 4x + 3 = 0$, then its equation is......

The equations of the tangents to the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$ making an inclination of $30^\circ$ with the major axis are

If $\lim\limits_{x \to 1} \dfrac{\sin(3x^2 - 4x + 1) - x^2 + 1}{2x^3 - 7x^2 + ax + b} = -2$, then the quadratic equation having roots $a$ and $b$ is