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25 Nov 2018, 10:29
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if the straight line y = x + c is tangent to the circle (x-1)^2 + (y+2)^2 = 4, what is the maximum possible value for the constant c ?
(A) \(1 - \sqrt 2\)
(B) \(1 + \sqrt 2\)
(C) \(3 - 2 \sqrt 3\)
(D) \(-3 + 2 \sqrt 2\)
(E) None above
Source: https://www.gmath.net
(A) \(1 - \sqrt 2\)
(B) \(1 + \sqrt 2\)
(C) \(3 - 2 \sqrt 3\)
(D) \(-3 + 2 \sqrt 2\)
(E) None above
Source: https://www.gmath.net
26 Nov 2018, 12:54
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fskilnik
\(? = {c_{\max }} = c\)
Algebraic approach:
\(\left\{ \begin{gathered}\\
\,\left( 1 \right)\,\,{\left( {x - 1} \right)^2} + {\left( {y + 2} \right)^2} = 4 \hfill \\\\
\,\left( 2 \right)\,\,y = x + c \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)\,\,{\text{in}}\,\,\left( 1 \right)} \,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\,2{x^2} + 2\left( {c + 1} \right)x + {c^2} + 4c + 1 = 0\)
\({\text{tangency}}\,\,\, \Rightarrow \,\,\,0 = \Delta = {\left[ {2\left( {c + 1} \right)} \right]^2} - 4\left( 2 \right)\left( {{c^2} + 4c + 1} \right)\,\,\, = \,\,\, \ldots \,\,\, = \,\,\, - 4\left( {{c^2} + 6c + 1} \right)\)
\({c^2} + 6c + 1 = 0\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{Bhaskara}}} \,\,\,\,\,\left\{ \begin{gathered}\\
\,{c_1} = - 3 - 2\sqrt 2 \hfill \\\\
\,{c_2} = - 3 + 2\sqrt 2 \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,? = {c_2} = - 3 + 2\sqrt 2\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
30 Nov 2018, 19:55
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fskilnik
\(? = {c_{\max }} = c\,\,\,\,\left( {{\rm{figure}}} \right)\)
The alternate solution I present below is essentially geometric:
\(- 2 - c = 1 - 2\sqrt 2 \,\,\,\,\, \Rightarrow \,\,\,\,? = c = - 3 + 2\sqrt 2\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
