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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Official Answer and Stats are available only to registered users. Register/Login.

_________________

Encourage me by pressing the KUDOS if you find my post to be helpful.

Help me win "The One Thing You Wish You Knew - GMAT Club Contest" http://gmatclub.com/forum/the-one-thing-you-wish-you-knew-gmat-club-contest-140358.html#p1130989

Last edited by Bunuel on 22 Aug 2014, 20:31, edited 1 time in total.

Even if you don't know the formula for the distance between two parallel lines, you can figure it out.

The two lines are parallel, having the same slope 2. A circle tangent to both lines will have the diameter equal to the distance between the two lines. To find this distance, take the perpendicular from the origin to each of the lines. The origin and the x-intercept and the y-intercept of the line \(y = 2x - 10\) form a right triangle with legs 5 and 10 and hypotenuse \(5\sqrt{5}\). Therefore, the height corresponding to the hypotenuse is \(10\cdot{5}/5\sqrt{5}=2\sqrt{5}\) (use the formula height = leg*leg/hypotenuse). The distance between the origin and the line \(y = 2x + 5\) is half of the distance between the origin and the line \(y = 2x - 10\), because the x-intercept and the y-intercept of the line \(y = 2x + 5\) are -5/2 and 5, which with the origin, form a right triangle similar to the right triangle discussed above.

So, the diameter of the circle is \(3\sqrt{5}\) and the area of the circle is \(\pi\cdot9\cdot{5}/4=(45/4)\pi\).
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

09 Oct 2012, 07:09

2

This post received KUDOS

My answer by using first principles and not the A and B formula

Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines

The first thing to use is the perpendicular bi-sector between the two lines. If y=2x is the gradient of the lines, then y= -1/2x is the gradient of the bisector.

Step 1, solve the two simultaneous equations in order to find the vectors of the line you seek.

y=2x+5 y= -1/2x x = -2 Subbing -2 into y=2x+5=1 So co-ord 1 is (-2,1)

Next y=2x-10 y=-1/2x x=4 Subbing 4 into y=2x-10=-2 Co-ord to is (4,-2)

Now you have a pythagorus equation Triangle base = 6 Triangle height = 3

Hypotenuse (the diameter of the circle) ^2 = 36 + 9 Hyp= diameter = sqrt(45) Radius = Sqrt(45)/2 Area = Pi x r^2 =45/4 x pi
_________________

If you find my post helpful, please GIVE ME SOME KUDOS!

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

15 Oct 2012, 09:44

nktdotgupta I have tried to solve with your way, but when I try to calculate d I receive another result: \(d = \frac{15}{\sqrt{5}}\) Then I divide d with 2 for find r and eventually I elevate r^2 and moltiplicate with pie, but the result is different. I'm not able to find the error.

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

15 Oct 2012, 10:03

2

This post received KUDOS

IanSolo wrote:

nktdotgupta I have tried to solve with your way, but when I try to calculate d I receive another result: \(d = \frac{15}{\sqrt{5}}\) Then I divide d with 2 for find r and eventually I elevate r^2 and moltiplicate with pie, but the result is different. I'm not able to find the error.

\(\frac{15}{\sqrt{5}}=\frac{15\sqrt{5}}{\sqrt{5}\sqrt{5}}=\frac{15\sqrt{5}}{5}=3\sqrt{5}\) which is the same as \(\sqrt{9\cdot{5}}=\sqrt{45}\).
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

16 Oct 2012, 05:58

IanSolo wrote:

nktdotgupta I have tried to solve with your way, but when I try to calculate d I receive another result: \(d = \frac{15}{\sqrt{5}}\) Then I divide d with 2 for find r and eventually I elevate r^2 and moltiplicate with pie, but the result is different. I'm not able to find the error.

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

22 Aug 2014, 14:30

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

23 Aug 2014, 10:20

Point (0,5) satisfies equation y=2x+5 Point (5,0) satisfies equation y=2x-10 Take (0,0) to form a right triangle with points (0,5) and (5,0) Height of triangle= distance between (0,0) and (0,5) = 5 Base of triangle=distance between (0,0) and (5,0) = 5 Therefore hypotenuse =5\sqrt{2} That is the diameter of the circle. Area of circle= pi*r^2 =pi*(5\sqrt{2}/2)^2 =pi*50/4

Can anyone explain what am I missing in this approach.

Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines

Point (0,5) satisfies equation y=2x+5 Point (5,0) satisfies equation y=2x-10 Take (0,0) to form a right triangle with points (0,5) and (5,0) Height of triangle= distance between (0,0) and (0,5) = 5 Base of triangle=distance between (0,0) and (5,0) = 5 Therefore hypotenuse =5\sqrt{2} That is the diameter of the circle. Area of circle= pi*r^2 =pi*(5\sqrt{2}/2)^2 =pi*50/4

Can anyone explain what am I missing in this approach.

The point is that a circle tangent to both those lines must have the diameter equal to the distance between the two lines. The distance between two parallel lines is the length of perpendicular from one to another, while line segment joining (5,0) and (0,5) is NOT perpendicular to the lines:

Attachment:

Untitled.png [ 12.54 KiB | Viewed 3764 times ]

So, this line segment cannot be the diameter of the circle.

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

23 Aug 2014, 12:00

Bunuel wrote:

desaichinmay22 wrote:

Given the two lines y = 2x + 5 and y = 2x - 10, what is the area of the largest circle that can be inscribed such that it is tangent to both lines

Point (0,5) satisfies equation y=2x+5 Point (5,0) satisfies equation y=2x-10 Take (0,0) to form a right triangle with points (0,5) and (5,0) Height of triangle= distance between (0,0) and (0,5) = 5 Base of triangle=distance between (0,0) and (5,0) = 5 Therefore hypotenuse =5\sqrt{2} That is the diameter of the circle. Area of circle= pi*r^2 =pi*(5\sqrt{2}/2)^2 =pi*50/4

Can anyone explain what am I missing in this approach.

The point is that a circle tangent to both those lines must have the diameter equal to the distance between the two lines. The distance between two parallel lines is the length of perpendicular from one to another, while line segment joining (5,0) and (0,5) is NOT perpendicular to the lines:

Attachment:

Untitled.png

So, this line segment cannot be the diameter of the circle.

Hope it's clear.

Hi Bunuel,

I got your point. Can you please suggest how to derive the length of perpendicular using graphical approach. I am not able to get it.

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

24 Aug 2014, 03:50

I used the following formula to find distance between lines: |ax+by+c|/sqrt(a^2+b^2), got 15/sqrt5, halved and got 15/2*(sqrt5), squared it to get 45/4pi

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

24 Aug 2014, 06:21

3

This post was BOOKMARKED

Graphical approach:

Straight lines with equation f(x) = 2x + 5 and g(x) = 2x - 10 are parallel; slope of each line equal to 2.

Co-ordinate of y-intercepts of lines f(x) and g(x) are (0,5) and (0,-10)

Equation of the line perpendicular to f(x) and g(x) and passing through y-intercept of g(x):h(x) = -(1/2)x-10

Point of intersection between h(x) and f(x) : (-6,-7) [ Point of intersection satisfies both line equations f(x) and h(x). Thus, \(2x+5=\frac{-1}{2}x-10\) or\(x =-6\) and \(y=-7\)]

Distance between parallel lines = Distance between (-6,-7) and (0,-10) = \(\sqrt{( -10+7)^2+(0+6)^2}\)\(=3\sqrt{5}\)

Largest circle that can be inscribed between lines f(x) and g(x) must have the diameter equal to the distance between these two lines = \(3\sqrt{5}\)

Area of the circle = \(\pi*(3\sqrt{5}/2)^2= \frac{45}{4}*\pi\)

Answer: (A)

Attachments

Distance-Between-Parallel-Lines.png [ 24.44 KiB | Viewed 3623 times ]

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

28 Aug 2014, 05:59

Temurkhon wrote:

I used the following formula to find distance between lines: |ax+by+c|/sqrt(a^2+b^2), got 15/sqrt5, halved and got 15/2*(sqrt5), squared it to get 45/4pi

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

05 Dec 2015, 11:03

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the [#permalink]

Show Tags

26 Mar 2016, 17:35

wow..this problem is a nightmare..i guessed it..but tried to solve it for more than half an hour... I picked A, because it is written in X/4 pi. I knew that one point from one line to the other point on the other line would form the diameter. to find the radius, we need to divide it by 2. the area would be (d/2)^2 or d^2 / 4 I did not know how to solve, so picked A, as it had /4...

gmatclubot

Re: Given the two lines y = 2x + 5 and y = 2x - 10, what is the
[#permalink]
26 Mar 2016, 17:35

Campus visits play a crucial role in the MBA application process. It’s one thing to be passionate about one school but another to actually visit the campus, talk...

Its been long time coming. I have always been passionate about poetry. It’s my way of expressing my feelings and emotions. And i feel a person can convey...

Marty Cagan is founding partner of the Silicon Valley Product Group, a consulting firm that helps companies with their product strategy. Prior to that he held product roles at...

Written by Scottish historian Niall Ferguson , the book is subtitled “A Financial History of the World”. There is also a long documentary of the same name that the...