• Get 300+ Practice Questions
25 Video lessons and 6 Webinars for FREE

Available with Beat the GMAT members only code

• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

• FREE GMAT Exam
Know how you'd score today for $0 Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

Available with Beat the GMAT members only code

Point P is the point of the circle x^2 + y^2 -2x -4y = 4 wit

tagged by: fskilnik

00:00

A

B

C

D

E

Global Stats

Difficult

[GMATH practice question]

Point P is the point satisfying x^2 + y^2 -2x -4y = 4 with maximum possible vertical coordinate. What is the sum of the coordinates of P?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7

Answer: __(C)____

_________________
Fabio Skilnik :: www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

Last edited by fskilnik on Fri Sep 14, 2018 8:50 am; edited 1 time in total

GMAT/MBA Expert

GMAT Instructor
Joined
09 Oct 2010
Posted:
523 messages
Followed by:
25 members
Upvotes:
59
fskilnik wrote:
[GMATH practice question]

Point P is the point satisfying x^2 + y^2 -2x -4y = 4 with maximum possible vertical coordinate. What is the sum of the coordinates of P?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7
$P = \left( {{x_P}\,,\,\,{y_P}} \right)\,\,\, \in \,\,\,\,\left\{ {\,\left( {x,y} \right)\,\,\,:\,\,\,{x^2} - 2x + {y^2} - 4y = 4\,} \right\}$
${y_P}\,\,\max \,\,\,,\,\,\,\,\,? = {x_P} + {y_P}$

Let´s apply the "filling the squares" technique presented in our course!

${x^2} - 2x + {y^2} - 4y = 4\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\underbrace {{x^2} - 2x + \underline 1 }_{{{\left( {x - 1} \right)}^{\,2}}} + \underbrace {{y^2} - 4y + \underline 4 }_{{{\left( {y - 2} \right)}^{\,2}}} = \underbrace {4 + \underline 1 + \underline 4 }_9$
$P\,\, \in \,\,\,\left\{ {\,\,\left( {x,y} \right)\,\,:\,\,\,{{\left( {x - 1} \right)}^2} + {{\left( {y - 2} \right)}^2} = {3^2}} \right\}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,P\,\, \in \,\,\,\, \odot \,\,\left\{ \begin{gathered} \,{\text{Centre}}\, = \left( {1,2} \right) \hfill \\ {\text{Radius}} = 3 \hfill \\ \end{gathered} \right.$
$\left. \begin{gathered} P = \left( {{x_P}\,,\,\,{y_P}} \right)\,\, \in \,\,\,\, \odot \,\, \hfill \\ {y_P}\,\,\max \,\, \hfill \\ \end{gathered} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{geometrically}}\,\,{\text{evident}}\,!} \,\,\,\,\,P = \left( {1,2 + 3} \right) = \left( {1,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 6$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

Last edited by fskilnik on Sat Sep 15, 2018 4:00 pm; edited 2 times in total

Top Member

Master | Next Rank: 500 Posts
Joined
15 Oct 2009
Posted:
308 messages
Upvotes:
27
fskilnik wrote:
[GMATH practice question]

Point P is the point satisfying x^2 + y^2 -2x -4y = 4 with maximum possible vertical coordinate. What is the sum of the coordinates of P?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7

Answer: __(C)____
Reorganizing the problem: y^2 - 4y +x^2- 2x = 4

To maximize the vertical is to maximize y. To maximize y, we need to minimize the expression x^2 - 2x since to the extent it is positive will reduce the right side, 4, available for the y expression to satisfy.

Looking at x^2 - 2x it is tempting to believe the minimum value is 0, but can we do better than that ?

If we set x=1, then the expression reduces to -1, that's better than 0. If we set x=2, the expression = 0, so headed in the wrong direction.

If we set x = 0, the expression = 0 , again the wrong direction. Testing x=-1 the expression = 3.

So x = 1 minimizes the expression. Plugging x=1 into the expression and solving for y:

y^2-4y + 1 - 2 = 4 > y^2-4y - 5 =0

Solving for Y: (y+1)(y-5) = 0 so y = -1, 5. The maximum is 5.

So the coordinates of the point P where y is a maximum, 5, are (1,5) so the sum of the coordinates is 6, C

GMAT/MBA Expert

GMAT Instructor
Joined
09 Oct 2010
Posted:
523 messages
Followed by:
25 members
Upvotes:
59
fskilnik wrote:
[GMATH practice question]

Point P is the point satisfying x^2 + y^2 -2x -4y = 4 with maximum possible vertical coordinate. What is the sum of the coordinates of P?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7
Alternate solution (adapted from the very nice idea of regor60, posted above. Thank you for your contribution!):

$P = \left( {{x_P}\,,\,\,{y_P}} \right)\,\,\, \in \,\,\,\,\left\{ {\,\left( {x,y} \right)\,\,\,:\,\,\,{x^2} - 2x + {y^2} - 4y = 4\,} \right\}$
${y_P}\,\,\max \,\,\,,\,\,\,\,\,? = {x_P} + {y_P}$
${x^2} - 2x + {y^2} - 4y = 4\,\,\,\,\, \Leftrightarrow \,\,\,\,{y^2} - 4y = 4 - \left( {{x^2} - 2x + \underline 1 } \right) + \underline 1 = 5 - {\left( {x - 1} \right)^2}$
${y^2} - 4y = 5 - {\left( {x - 1} \right)^2} \leqslant 5$
$y\,\,\max \,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ {\begin{array}{*{20}{c}} {x = {x_p} = 1} \\ {{y_p}^2 - 4{y_p} = 5} \end{array}\begin{array}{*{20}{c}} {} \\ {\,\,\,\mathop \Rightarrow \limits^{S = 4\,,\,P = - 5} \,\,\,\,{y_p} = \max \left\{ {5, - 1} \right\}\,\, = 5\,\,\,\,} \end{array}} \right.$
$? = 1 + 5 = 6$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

Last edited by fskilnik on Fri Sep 14, 2018 12:58 pm; edited 1 time in total

GMAT/MBA Expert

GMAT Instructor
Joined
25 May 2010
Posted:
14733 messages
Followed by:
1849 members
Upvotes:
13060
GMAT Score:
790
fskilnik wrote:
[GMATH practice question]

Point P is the point satisfying x^2 + y^2 -2x -4y = 4 with maximum possible vertical coordinate. What is the sum of the coordinates of P?

(A) 5
(B) 5.5
(C) 6
(D) 6.5
(E) 7
(x-h)² + (y-k)² = r² is a circle with a center at (h, k) and a radius of r.

x² + y² - 2x - 4y = 4

x² - 2x + y² - 4y = 4

x² - 2x + 1 + y² - 4y + 4 = 4 + 1 + 4

(x-1)² + (y-2)² = 9

(x-1)² + (y-2)² = 3³

The equation above constitutes a circle with a center at (1, 2) and a radius of 3.
Since r=3, the highest point is located exactly 3 units above the center:
As shown in the figure above, the highest point is at (1, 5).
Sum of the coordinates = 1+5 = 6.

The correct answer is C.

_________________
Mitch Hunt
Private Tutor for the GMAT and GRE
GMATGuruNY@gmail.com

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.
Student Review #1
Student Review #2
Student Review #3

Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

Top First Responders*

1 Jay@ManhattanReview 83 first replies
2 Brent@GMATPrepNow 68 first replies
3 fskilnik 55 first replies
4 GMATGuruNY 36 first replies
5 ceilidh.erickson 13 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

Most Active Experts

1 fskilnik

GMAT Teacher

199 posts
2 Brent@GMATPrepNow

GMAT Prep Now Teacher

160 posts
3 Scott@TargetTestPrep

Target Test Prep

109 posts
4 Jay@ManhattanReview

Manhattan Review

95 posts
5 GMATGuruNY

The Princeton Review Teacher

90 posts
See More Top Beat The GMAT Experts