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JEE Main 202117 MarchMorning Shift
Exam
JEE Main
Year
2021
Date
17 March
Shift
Morning Shift
Subject
Mathematics
Chapter
Circle

JEE Main 2021 · 17 March · Morning Shift · Mathematics · Circle

The line 2x - y + 1 = 0 is a tangent to the circle at the point (2,...

The line 2x−y+1=02x - y + 1 = 0 is a tangent to the circle at the point (2,5)\left(2, 5\right) and the centre of the circle lies on x−2y=4.x - 2y = 4 . Then, the radius of the circle is:The line 2x - y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x - 2y = 4

Options

  1. A353\sqrt{5}
  2. B535\sqrt{3}
  3. C545\sqrt{4}
  4. D454\sqrt{5}

Correct answer

A. 353\sqrt{5}

Step-by-step solution

Given

The line 2x - y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x - 2y = 4

Slope of OA=5−(h−4)22−h=m1OA = \dfrac{5 - \dfrac{\left(h - 4\right)}{2}}{2 - h} = {m}_{1}Slope of the given tangent equation=m2=2= {m}_{2} = 2

⇒(5−(h−4)22−h)(2)=−1⇒ \left(\dfrac{5 - \dfrac{( h - 4 )}{2}}{2 - h}\right)\left(2\right) = - 1

⇒h=8⇒ h = 8

Center=(8,2)= \left(8, 2\right)

Radius=(8−2)2+(2−5)2=35= \sqrt{( 8 - 2{)}^{2} + ( 2 - 5{)}^{2}} = 3\sqrt{5}

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