Circle

If the line $\frac{x}{a}+\frac{y}{b}-1$ moves such that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$, then the locus of the foot of the perpendicular from the origin io the line is

The values of $\alpha$ for which the point $(\alpha-1, \alpha+1)$ lies in the larger segment of the circle $x^2+y^2-x-y-6=0$ made by the chord whose equation is $x+y-2=0$ is

The locus of the mid-points of the chord of the circle $x^2+y^2=4$ which subtends a right angle at the origin, is

From a point $A$ common tangents are drawn to the circle $x^2+y^2=a^2 / 2$ and parabola $y^2=4 a x$. The area of the quadrilateral formed by the common tangents to the chord of contact of the parabola is

The length of the common chord of the two circles $(x-a)^2+(y-b)^2=c^2$ and $(x-b)^2+(y-a)^2=c^2$ is

A circle of radius 2 is touching both the axes and a circle with centre $(6,5)$. The distance between their centres is

The number of common tangents to two circles $x^2+y^2=4$ and $x^2+y^2-8 x+12=0$ is

Consider four circles $(x \pm 1)^2+(y \pm 1)^2=1$, then the equation of smaller circle touching these four circle is

A rhombus is inscribed in the region common to the two circles $x^2+y^2-4 x-12=0$ and $x^2+y^2+4 x-12=0$ with two of its vertices on the line joining the centres of the circles. The area of rhombus is

The equation of circle which passes through the origin and cuts off intercepts 5 and 6 from the positive parts of the axes respectively, is $\left(x-\frac{5}{2}\right)^2+(y-3)^2=\lambda$, where $\lambda$ is

The locus of the point of intersection of two perpendicular tangents to a circle is called

Equation of tangent to the circle $x^2+y^2-2 x-2 y+1=0$ perpendicular to $y=x$ is given by