Circles · JEE Advanced previous year questions

Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1=1+2i$ and $z_2=3i$ be two complex numbers, where $i=\sqrt{-1}$. Let $$S=\{(x,y)\in \mathbb{R}\times \mathbb{R}:|x+iy-z_1|=2|x+iy-z_2|\}.$$ Then which of the following statements is (are) TRUE?

1. A.$S$ is a circle with centre $\left(-\frac{1}{3},\frac{10}{3}\right)$
2. B.$S$ is a circle with centre $\left(\frac{1}{3},\frac{8}{3}\right)$
3. C.$S$ is a circle with radius $\frac{\sqrt{2}}{3}$
4. D.$S$ is a circle with radius $\frac{2\sqrt{2}}{3}$

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two distinct points on the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ such that $y_1>0$, and $y_2>0$. Let $C$ denote the circle $x^2+y^2=9$, and $M$ be the point $(3,0)$. Suppose the line $x=x_1$ intersects $C$ at $R$, and the line $x=x_2$ intersects $C$ at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle ROM=\frac{\pi }{6}$ and $\angle SOM=\frac{\pi }{3}$, where $O$ denotes the origin $(0,0)$. Let $|XY|$ denote the length of the line segment $XY$. Then which of the following statements is (are) TRUE?

1. A.The equation of the line joining $P$ and $Q$ is $2x+3y=3(1+\sqrt{3})$
2. B.The equation of the line joining $P$ and $Q$ is $2x+y=3(1+\sqrt{3})$
3. C.If $N_2=(x_2,0)$, then $3|N_{2}Q|=2|N_{2}S|$
4. D.If $N_1=(x_1,0)$, then $9|N_{1}P|=4|N_{1}R|$

Let the straight line $y=2x$ touch a circle with center $(0,\alpha )$, $\alpha >0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1{B}_1$ is a diameter of the circle. Let $\alpha +r=5+\sqrt{5}$. Match each entry in List-I to the correct entry in List-II. List-I: (P) $\alpha$ equals (Q) $r$ equals (R) $A_1$ equals (S) $B_1$ equals List-II: (1) $(-2,4)$ (2) $\sqrt{5}$ (3) $(-2,6)$ (4) 5 (5) $(2,4)$ The correct option is

1. A.(P) $\to$ (4) \quad (Q) $\to$ (2) \quad (R) $\to$ (1) \quad (S) $\to$ (3)
2. B.(P) $\to$ (2) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (3)
3. C.(P) $\to$ (4) \quad (Q) $\to$ (2) \quad (R) $\to$ (5) \quad (S) $\to$ (3)
4. D.(P) $\to$ (2) \quad (Q) $\to$ (4) \quad (R) $\to$ (3) \quad (S) $\to$ (5)

Let $A_1,A_2,A_3,\dots ,A_8$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P{A}_i$ denote the distance between the points $P$ and $A_i$ for $i=1,2,\dots ,8$. If $P$ varies over the circle, then the maximum value of the product $P{A}_1\cdot P{A}_2\cdots P{A}_8$, is ____.

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1<r<3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB=\sqrt{5}$, then the value of $r^2$ is ____.

PARAGRAPH “I” Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. (There are two questions based on PARAGRAPH “I”, the question given below is one of them) Let $a$ be the area of the triangle $ABC$. Then the value of $(64a{)}^2$ is ____.

PARAGRAPH “I” Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1. (There are two questions based on PARAGRAPH “I”, the question given below is one of them) Then the inradius of the triangle $ABC$ is ____.