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JEE Main 202529 JanuaryEvening Shift
Exam
JEE Main
Year
2025
Date
29 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Circle

JEE Main 2025 · 29 January · Evening Shift · Mathematics · Circle

Let a circle C pass through the points (4,2) and (0,2) , and its...

Let a circle CC pass through the points (4,2)(4,2) and (0,2)(0,2), and its centre lie on 3x+2y+2=03 x+2 y+2=0. Then the length of the chord, of the circle CC, whose mid-point is (1,2)(1,2), is :

Options

  1. A3\sqrt{3}
  2. B222 \sqrt{2}
  3. C232 \sqrt{3}
  4. D424 \sqrt{2}

Correct answer

C. 232 \sqrt{3}

Step-by-step solution

Let the centre be (−2a,6a−22)≡(−2a,3a−1)\left(-2 a, \frac{6 a-2}{2}\right) \equiv(-2 a, 3 a-1) Centre is equal distance from (4,2)and(0,2)(4,2) and (0,2)⇒(4+2a)2+(3a−3)2=(−2a−0)2+(3a−3)2⇒(2a+4)2+9(a−1)2=4a2+9(a−1)2⇒4a2+16+16a=4a2⇒a=−1⇒ centre ≡(2,−4)⇒ Radius =40\begin{aligned} & \Rightarrow \sqrt{(4+2 a)^2+(3 a-3)^2}=\sqrt{(-2 a-0)^2+(3 a-3)^2} \\ & \Rightarrow(2 a+4)^2+9(a-1)^2=4 a^2+9(a-1)^2 \\ & \Rightarrow 4 a^2+16+16 a=4 a^2 \Rightarrow a=-1 \\ & \Rightarrow \text{ centre } \equiv(2,-4) \Rightarrow \text{ Radius }=\sqrt{40}\end{aligned}⇒AM2=(40)2−(37)2⇒2AM=AB=23\begin{aligned} & \Rightarrow A M^2=(\sqrt{40})^2-(\sqrt{37})^2 \\ & \Rightarrow 2 A M=A B=2 \sqrt{3}\end{aligned}$$
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