Embibe Experts Solutions for Chapter: Circle, Exercise 1: Exercise 1

A circle of constant radius $r$ passes through the origin $O,$ and cuts the axes at $A$ and $B.$ The locus of the foot of the perpendicular from $O$ to $AB$ is ${\left(x^2+y^2\right)}^k=4r^2{x}^2{y}^2,$ then the value of $k$ is

Six points $\left(x_i,y_i\right), i=1, 2\dots \dots \dots , 6$ are taken on the circle $x^2+y^2=4$ such that $\sum _{i=1}^6{x}_i=8$ and $\sum _{i=1}^6{y}_i=4$, the line segment joining orthocentre of a triangle made by any three points and the centroid of the triangle made by other three points passes through a fixed point $(h,k),$ then $h+k$ is

A triangle has two of its sides along the axes, its third side touches the circle $x^2+y^2-4x-4y+4=0$ locus of circumcentre of triangle is

A circle cutting the circle $x^2+y^2=4$ orthogonally and having its centre on the line $2x-2y+9=0,$ passes through two fixed points. These points are

The straight line $lx+my=1$ intersects $p{x}^2+2qxy+r{y}^2=s$ at $AB.$ Chord $AB$ subtends a right angle at the origin. If $lx+my=1$ is a tangent to a circle $x^2+y^2=a^2,$ then $a=$

Tangents and normal are drawn from a point $P(3, 1)$ to circle $C$ whose equation is $x^2+y^2-2x-2y+1=0.$ Let the points of contact of tangents be $T_i$ $\left(x_i, y_i\right),$ where $i=1, 2$ and feet of normals be $N_1$ and $N_2\left(N_1\right.$ is near $\left.P\right).$ Tangents are drawn at $N_1$ and $N_2$ and normals are drawn at $T_1$ and $T_2$

The point of contact of the tangent to the circle $x^2+y^2=5$ at the point $\left(1, -2\right)$ which touches the circle $x^2+y^2-8x+6y+20=0$, is $\left(h, k\right)$ then $\left(2h^2+3k^2\right)$ is equal to

If the length of the tangent from any point on the circle $(x-3{)}^2+(y+2{)}^2=5r^2$ to the circle $(x-3{)}^2+(y+2{)}^2=r^2$ is $16$ units, then the area between the two circles is $\lambda \pi$ sq. units then $\lambda$ is equal to