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16 Sep 2014, 01:01
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Is the line \(y = kx + b\) tangent to the circle defined by \(x^2 + y^2 = 1\)?
(1) \(k + b = 1\)
(2) \(k^2 + b^2 = 1\)
(1) \(k + b = 1\)
(2) \(k^2 + b^2 = 1\)
16 Sep 2014, 01:01
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Official Solution:
Is the line \(y = kx + b\) tangent to the circle defined by \(x^2 + y^2 = 1\)?
Note that a circle described by the equation \(x^2 + y^2 = 1\) is centered at the origin with a radius of \(r = 1\).
(1) \(k+b=1\). If \(k = 0\) and \(b = 1\), then the equation of the line becomes \(y=1\) and this line is tangent to the circle but if \(k=1\) and \(b=0\), then the equation of the line becomes \(y=x\) and this line is NOT tangent to the circle. Not sufficient.
(2) \(k^2+b^2=1\).
The same example is valid for this statement too. Not sufficient.
(1)+(2) Again the same example satisfies both statements: if \(k=0\) and \(b=1\), then the equation of the line becomes \(y=1\) and this line is tangent to the circle but if \(k=1\) and \(b=0\), then the equation of the line becomes \(y=x\) and this line is NOT tangent to the circle. Not sufficient. Look at the diagram below to see both cases:
Answer: E
Is the line \(y = kx + b\) tangent to the circle defined by \(x^2 + y^2 = 1\)?
Note that a circle described by the equation \(x^2 + y^2 = 1\) is centered at the origin with a radius of \(r = 1\).
(1) \(k+b=1\). If \(k = 0\) and \(b = 1\), then the equation of the line becomes \(y=1\) and this line is tangent to the circle but if \(k=1\) and \(b=0\), then the equation of the line becomes \(y=x\) and this line is NOT tangent to the circle. Not sufficient.
(2) \(k^2+b^2=1\).
The same example is valid for this statement too. Not sufficient.
(1)+(2) Again the same example satisfies both statements: if \(k=0\) and \(b=1\), then the equation of the line becomes \(y=1\) and this line is tangent to the circle but if \(k=1\) and \(b=0\), then the equation of the line becomes \(y=x\) and this line is NOT tangent to the circle. Not sufficient. Look at the diagram below to see both cases:
Answer: E
General Discussion
11 Jul 2016, 07:18
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I think this is a high-quality question and I agree with explanation.
