The straight line x+2y=1 meets the coordinate axes at A and B. A circle is drawn through A,B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is :
A
4√5
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B
√54
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C
2√5
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D
√52
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Solution
The correct option is D√52
Given line AB is x+2y=1
Slope of line AB=−12
Co-ordinates of A and B are A=(1,0),B=(0,12)
Co-ordinates of C=⎛⎜
⎜
⎜⎝1+02,0+122⎞⎟
⎟
⎟⎠=(12,14)
Slope of OC=0−140−12=12
∴ Slope of tangent =−2
So, the equation of tangent is 2x+y=0
Sum of distances from A(1,0) and B(0,12) of tangent =∣∣
∣∣2√22+12∣∣
∣∣+∣∣
∣
∣∣12√22+12∣∣
∣
∣∣=2√5+12√5=52√5=√52