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if the straight line y = x + c is tangent to the circle (x-1)^2 + (y

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Posts: 935
if the straight line y = x + c is tangent to the circle (x-1)^2 + (y  [#permalink]

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25 Nov 2018, 10:29
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Difficulty:

95% (hard)

Question Stats:

22% (02:12) correct 78% (02:19) wrong based on 37 sessions

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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

_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
if the straight line y = x + c is tangent to the circle (x-1)^2 + (y  [#permalink]

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26 Nov 2018, 12:54
fskilnik wrote:
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

$$? = {c_{\max }} = c$$

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.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: if the straight line y = x + c is tangent to the circle (x-1)^2 + (y  [#permalink]

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30 Nov 2018, 19:55
fskilnik wrote:
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

$$? = {c_{\max }} = c\,\,\,\,\left( {{\rm{figure}}} \right)$$

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.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: if the straight line y = x + c is tangent to the circle (x-1)^2 + (y   [#permalink] 30 Nov 2018, 19:55
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