Circles · JEE Advanced previous year questions

2025 2 questions

1. Q72025 · 18 May 2025 · Paper 1· One or more correct

   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 $\frac{2\sqrt{2}}{3}$
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   Answer: A, D

2. Q72025 · 18 May 2025 · Paper 2· One or more correct

   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 $9|N_{1}P|=4|N_{1}R|$
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   Answer: A, C