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Question

# If the line 2x−y+1=0 is a tangent to the circle S≡0 at P(2,5) and the centre of the circle lies on x−2y=4, then the intercept made by the circle S≡0 on the x−axis is equal to

A
223
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B
241
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C
235
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D
263
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Solution

## The correct option is B 2√41Equation of normal at P is x+2y+k=0 ⇒2+2⋅5+k=0⇒k=−12 ∴ Equation of normal at P is x+2y−12=0 ⋯(1) Centre lies on x−2y−4=0 ⋯(2) Solving (1) and (2), we get centre, C≡(8,2) Radius, r=CP=√45 ∴ Equation of the circle is (x−8)2+(y−2)2=45 ⇒x2+y2−16x−4y+23=0 Length of x−intercept =2√g2−c=2√41

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