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Question

If the line y=3x+k touches the circle x2+y2=16, then find the value of k. 


Solution

Consider the given equation of a circle as x2+y2=16
Centre is (0, 0) and radius =4 as shown figure.

AB be a line passing through centre of circle. 
Tangent y=3x+k touches the circle at B(a,b)
a2+b2=16
AB is perpendicular to tangent.
Slope of AB=13
Equation of AB is
y=13× [AB passes through centre (0, 0)] 
b=13a     (2)
Substituting (2) in (1), we get,
a2+13a2=164a23=16a=±3b=±2
B(a,b) is on y=(23)+k±2=±6+kk=±8

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