Circles · JEE Advanced previous year questions

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2023 4 questions

Q112023 · 4 Jun 2023 · Paper 2· Numerical answer
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 $P A_{1} \cdot P A_{2} \dots P A_{8}$ , is ____.
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Answer: 512
Q132023 · 4 Jun 2023 · Paper 2· Numerical answer
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 $P Q$ and $S T$ of $C_{1}$ and $C_{2}$ are drawn. The tangent $P Q$ touches $C_{1}$ at $P$ and $C_{2}$ at $Q$ . The tangent $S T$ touches $C_{1}$ at $S$ and $C_{2}$ at $T$ . Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$ -axis at a point $B$ . If $A B = 5$ , then the value of $r^{2}$ is ____.
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Answer: 2
Q142023 · 4 Jun 2023 · Paper 2· Numerical answer
PARAGRAPH “I” Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{π}{2}$ 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 $A B C$ . Then the value of $(64 a)^{2}$ is ____.
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Answer: 1007.99 to 1008.01
Q152023 · 4 Jun 2023 · Paper 2· Numerical answer
PARAGRAPH “I” Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{π}{2}$ 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 $A B C$ is ____.
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Answer: 0.24 to 0.26