Embibe Experts Solutions for Chapter: Circle, Exercise 1: JEE Main - 16th April 2018

The locus of the middle points of the chords of the circle $C_1: {\left(x-4\right)}^2+{\left(y-5\right)}^2=4$ which subtend an angle ${\theta }_i$ at the centre of the circle $C_i$, is a circle of radius $r_i$. If ${\theta }_1=\frac{\mathrm{\pi }}{3}$, ${\theta }_3=\frac{2\mathrm{\pi }}{3}$ and ${r_1}^2={r_2}^2+{r_3}^2$, then ${\theta }_2$ is equal to

The points of intersection of the line $ax + by=0$, $(\mathrm{a}\ne \mathrm{b})$ and the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}=0$ are $A(\alpha ,0)$ and $B(1,\beta )$. The image of the circle with $AB$ as a diameter in the line $\mathrm{x}+\mathrm{y}+2=0$ is :

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^2+y^2-3x+10y-15=0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to

A circle with centre $(2,3)$and radius $4$ intersects the line $x+y=3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S(\alpha ,\beta )$, then $4\alpha -7\beta$ is equal to

Let $y=x+2, 4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h{)}^2+(y-k{)}^2=r^2$. Then $h+k$ is equal to :

Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2},\sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to  __________

Let a circle $C_1$ be obtained on rolling the circle $x^2+y^2-4x-6y+11=0$ upwards $4$ units on the tangent $\mathrm{T}$ to it at the point $\left(3,2\right)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:

The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2},\frac{1-a}{2}\right)$ on the circle $2x^2+2y^2-(1+a)x-(1-a)y=0$, is equal to :