GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 22 May 2019, 02:54 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. ### Request Expert Reply # if the straight line y = x + c is tangent to the circle (x-1)^2 + (y

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

GMATH Teacher P
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]

### Show Tags 00:00

Difficulty:   95% (hard)

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

### HideShow timer Statistics

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 P
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]

### Show Tags

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 P
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]

### Show Tags

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
Display posts from previous: Sort by

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.

#### MBA Resources  