If $(\alpha, \beta)$ is a point on the circle whose centre is on the $x$-axis and which touches the line $x + y = 0$ at $(2, -2)$,then the greatest value of $\alpha$ is

Consider the curves $C_1: y^2=4x$ and $C_2: x^2+y^2-6x+1=0$. Assertion $(A)$: The common tangents to the curves $C_1$ and $C_2$ are orthogonal. Reason $(R)$: $x-y+1=0$ and $x+y+1=0$ are the common tangents to the curves $C_1$ and $C_2$.

If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2x-4y+\alpha=0$ at $A(4,2)$ and $B(x_1, y_1)$,then $PA \cdot PB$ is equal to

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is

If a variable point $(x, y)$ satisfies the equation $x^2 + y^2 - 8x - 6y + 9 = 0$,then the range of $\frac{y}{x}$ is

Let $a$ and $b$ be two non-zero real numbers. The equation $(ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0$ represents: