topspin360 wrote:can anyone show how to do this one?

thanks.

What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = (3/4)x − 3 ?

1.4
sqrt(2)
1.7
sqrt(3)
2.0

See the sketch attached. It is pretty self-explanatory except for a couple of things.
For the given line equation, we have, the x and y intercepts as
y=(3/4)x-3

When x=0, y=-3,
When y=0, x=4.

Then draw the perpendicular from the origin to the line segment joining A and B. Call this point C. We need to find OC.

We have the equation of the circle as x^2+y^2 = 1.
Before we go any further, here's a disclaimer. We can't be expected to solve this one in under 2 minutes, since we have no idea (as far as GMAT is concerned) what x^2+y^2 = 1 signifies.

For the purpose of solving this question, I am going to tell you what it means. It tells us that the circle is centered at (0,0) and has a radius of 1. Since we are not expected to know this, this question as written would not be a valid GMAT question. Anyway, since I do know what it means, I have solved the question anyway, assuming that we were told in the question stem itself that the circle is centered at (0,0) and has a radius of 1. With that in mind, and the attached figure in sight, let's continue.

Now we need to find CD. If we can find OC, we will find CD = OC-OD = OC - 1. Rest of the solution follows in the attached figure.

Let me know if this helps

eagleeye wrote:
Key takeaways from this question:

Equation of a standard circle centered at (a,b) with radius r is:
(x-a)^2 + (y-b)^2 = r^2

Distance of a line with equation ax+by+c=0 from a point (m,n) is:

|am+bn+c|/sqrt(a^2+b^2)

Let me know if this helps

I haven't seen these equations nor these types of problems in my GMAT prep thus far. Should they be known or is there another approach? Thanks

alex.gellatly wrote:

eagleeye wrote:
Key takeaways from this question:

Equation of a standard circle centered at (a,b) with radius r is:
(x-a)^2 + (y-b)^2 = r^2

Distance of a line with equation ax+by+c=0 from a point (m,n) is:

|am+bn+c|/sqrt(a^2+b^2)

Let me know if this helps

I haven't seen these equations nor these types of problems in my GMAT prep thus far. Should they be known or is there another approach? Thanks

Good point. I started working on the alternative as soon as I came out of my Math Trance. See the GMAT version in the original post . You need to know what x^2+y^2 = 1 means to solve the question.

This equation is not covered in the GMAT math review under circles or co-ordinate geometry. Hence I would imagine, this exact question would never appear on the GMAT itself.

The best version that would work will say,

If a circle is centered at (0,0) and has radius of 1, what is the shortest distance between the circle and the line with the equation y=3x/4-3.

As the question is written right now, it is next to impossible to solve within the stipulated 2 minutes, unless you know what x^2+y^2 = 1 means, which as a GMAT person, you are not supposed to know.

Anyway, the solution above shows a couple of important points, which make this problem a helpful one.

Let me know if this helps

topspin360 wrote: What is the least possible distance between a point on the line y = (3/4)x − 3 and a point on the circle centered at the origin with a radius of 1?

1.4
sqrt(2)
1.7
sqrt(3)
2.0

I've changed the wording of the problem so that the solution does not require knowledge of the equation of a circle.

DRAW the figure:


The SHORTEST DISTANCE between a point and a line must form a RIGHT ANGLE with the line.
Thus:
AC = the shortest distance between the center of the circle and y = (3/4)x - 3.
EC = the shortest distance between a point on the circle and y = (3/4)x - 3.

∆ABD is a 3-4-5 triangle.
Area of ∆ABD = (1/2)(AD)(AB) = (1/2)(3)(4) = 6.
If we consider BD the base of ∆ABD, then AC is the corresponding height.
Since 1/2 the product of any base and height must yield the same area, we get:
(1/2)(BD)(AC) = 6
(1/2)(5)(AC) = 6.
AC = 12/5.

As the radius of the circle, AE = 1.
Thus, EC = AC - AE = 12/5 - 1 = 7/5 = 1.4.

The correct answer is A.

thanks all for each amazing explanations! really appreciate your effort in helping out. this problems came from one of the GMATclub tests... which are supposedly targeted at 750+ problems. But i guess some problems just go way too far.