Which of the following lines is a diameter of the circle $x^2 + y^2 - 6x - 8y - 9 = 0$?

If the chord joining the points $(1,2)$ and $(2,-1)$ on a circle subtends an angle of $\frac{\pi}{4}$ at any point on its circumference,then the equation of such a circle is:

Let $C$ be the circle with centre at $(1, 1)$ and radius $= 1$. If $T$ is the circle centred at $(0, y)$,passing through the origin and touching the circle $C$ externally,then the radius of $T$ is equal to:

Let the normals at all the points on a given curve pass through a fixed point $(a, b)$. If the curve passes through $(3, -3)$ and $(4, -2\sqrt{2})$,and given that $a - 2\sqrt{2}b = 3$,then $(a^{2} + b^{2} + ab)$ is equal to ..... .

Let a circle $C$ pass through the points $(4, 2)$ and $(0, 2)$,and its centre lie on the line $3x + 2y + 2 = 0$. Then the length of the chord of the circle $C$ whose midpoint is $(1, 2)$ is:

The equations of the circles,which touch both the axes and the line $4x+3y=12$ and have centers in the first quadrant,are