Circles

Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight lines x − y = 1 and 2x + y = 3 is :

If the normals of the parabola $y^2=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x-3)^2+(y+2)^2=r^2$, then the value of $r^2$ is

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta OPQ$ is $3\sqrt2$, then which of the following is (are) the coordinates of $P$?

Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $Py''+Qy'+1=0$, where $P$, $Q$ are functions of $x$, $y$ and $y'$ (here $y'=\dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$), then which of the following statements is (are) true?

Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x^2+(y-1)^2=2$. The straight line $x+y=3$ touches the curves $S$, $E_1$ and $E_2$ at $P$, $Q$ and $R$, respectively. Suppose that $PQ=PR=\dfrac{2\sqrt2}3$. If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$, respectively, then the correct expression(s) is(are)

Locus of the image of the point (2, 3) in the line $(2x-3y+4)+k(x-2y+3)=0$, $k\in\mathbb R$, is a :

The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$, is :

Let the tangents drawn to the circle, $x^2+y^2=16$ from the point P(0, h) meet the $x$-axis at points A and B. If the area of ΔAPB is minimum, then h is equal to :

If $y+3x=0$ is the equation of a chord of the circle, $x^2+y^2-30x=0$, then the equation of the circle with this chord as diameter is :

If the tangent to the conic, $y-6=x^2$ at (2, 10) touches the circle, $x^2+y^2+8x-2y=k$ (for some fixed k) at a point (α, β) ; then (α, β) is :

If the incentre of an equilateral triangle is (1, 1) and the equation of its one side is $3x+4y+3=0$, then the equation of the circumcircle of this triangle is :

If a circle passing through the point (−1, 0) touches $y$-axis at (0, 2), then the length of the chord of the circle along the $x$-axis is :

A circle $S$ passes through the point $(0,1)$ and is orthogonal to the circles $(x-1)^2+y^2=16$ and $x^2+y^2=1$. Then

The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P$, $Q$ and the parabola at the points $R$, $S$. Then the area of the quadrilateral $PQRS$ 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 origin and touching the circle C externally, then the radius of T is equal to :

If the differential equation representing the family of all circles touching $x$-axis at the origin is $\left(x^2-y^2\right)\dfrac{dy}{dx}=g(x)\,y$, then $g(x)$ equals :

If the point $(1,4)$ lies inside the circle $x^2+y^2-6x-10y+p=0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval :

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2+y^2-4x-4y+6=0$ and $x^2+y^2-10x-10y+\lambda=0$ is the interval :

Let $L_1$ be the length of the common chord of the curves $x^2+y^2=9$ and $y^2=8x$, and $L_2$ be the length of the latus rectum of $y^2=8x$, then :

For the two circles $x^2+y^2=16$ and $x^2+y^2-2y=0$, there is/are :