Verified Solution • 2025–26 Session moderate

QJEEM25Ma9e07: JEE Main 2025 Mathematics - Class 11 JEEM25Ma9e07 Solution: Let a circle C pass through the points (4, 2) and (0, 2), and…

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

Multiple Choice Options

  • A 4 2\sqrt{2}
  • B 2 2\sqrt{2}
  • C 2 3\sqrt{3}
  • D 3\sqrt{3}

Step-by-Step Verified Solution

JEE Main 2025 (Online) 29th January Evening Shift Mathematics - Circle Question 6 English Explanation

amp;amp;MAB=0⇒OM is vertical amp;⇒α=2amp;∴ Centre (0)≡(2,−4)amp;r=OA=(2−4)2+(2+4)2=40amp; mid point of chord is N≡(1,2)∴ON=37amp;∴ length of chord amp;=2r2−(ON)2amp;=240−37=23\begin{aligned}&\begin{aligned}& \mathrm{M}_{A B}=0 \Rightarrow \mathrm{OM} \text { is vertical } \\& \Rightarrow \alpha=2 \\& \therefore \text { Centre }(0) \equiv(2,-4) \\& \quad r=O A=\sqrt{(2-4)^2+(2+4)^2}=\sqrt{40}\end{aligned}\\&\text { mid point of chord is } \mathrm{N} \equiv(1,2) \quad \therefore \mathrm{ON}=\sqrt{37}\\&\begin{aligned}\therefore \text { length of chord } & =2 \sqrt{\mathrm{r}^2-(\mathrm{ON})^2} \\& =2 \sqrt{40-37}=2 \sqrt{3}\end{aligned}\end{aligned}

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