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Question

If ax+by5=0 is the equation of the shortest chord of the circle (x3)2+(y4)2=4 passing through the point (2,3), then |a+b| is

A
2
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B
4
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C
1
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D
0
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Solution

The correct option is D 2

Perpendicular is the shortest distance from any point to a line and the perpendicular from the circle's center will always bisect the cord

The shortest chord through a point is always perpendicular to the line(AC) joining the center and the point i.e. the point will be the midpoint of the chord

putting value of (2,3) in the equation of the circle we get that,
(23)2+(34)24=24=2<0
Hence, the point lies inside the circle

Slope of line joining the center and point is
4332=1

Slope of ax+by5=0 is 1
ab=1
a=b
Also, (2,3) lies on the chord ax+by5=0
2a+3b5=0
5a5=0 and 5b5=0 since a=b
a=b=1
|a+b|=1+1=2


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