Least possible distance?

[Chris Cross]

Hello,

Please help me understand how to solve the following problem quickly:

What is the least possible distance between a point on the circle x² + y² = 1 and a point on the line y = (3/4)x - 3?

[manpsingh87]

Hello,

Please help me understand how to solve the following problem quickly:

What is the least possible distance between a point on the circle x² + y² = 1 and a point on the line y = (3/4)x - 3?

x^2+y^2=1; represents circle having center (0,0) and radius=1,

now the line y=(3/4)x-3; intersects the x and y axis at the point (4,0) and (0,-3) respectively,if we plot the graph of the line and circle we will notice that least distance between the line and the point on the circle = (distance b/w the origin and line)-radius..!! (plot the graph you'll notice why i subtracted r from the distance b/w the origin and line.);

distance between the point (m,n) and line ax+by+c=0 can be found by using the distance formula..!!!
d=|am+bn+c|/sqrt(a^2+b^2)......1);

here (m,n)=(0,0)
a=-3/4,
b=1,
c=-3,

substituting above values in 1) we have;
|-3|/sqrt(9/16+1);
3/(5/4)=12/5;

hence required minimum distance=12/5-1;
7/5=1.4..!!!

i hope it helps...!!!

[clock60]

i got the same answer
but followed slightly easier way
here we have right tringle with sides 3,4,5 (it can easy be found) formed with line y=3/4x-3,line x, and line y.
the area of the triangle=1/2*3*4=6
but on the other hand the area=1/2*height*base, 6=1/2*heigth*5, height=12/5
and the least possible distance=12/5-1(radius of the circle)=12/5-5/5=7/5

[Chris Cross]

Thank you
That helped a lot!
I kinda forgot all that geometry stuff, have to get into it again in the next days.

Peace

[jaymw]

Where is this question from?

I have never seen anything like this on a GMAT exam! Some question ask about points in the coordinate plane and how far away they are from a certain line. I have, however, never encountered a question
that presupposes knowledge of circle and distance forumlae together.

I believe that for your GMAT studies you should not "waste" your time on such problems. They are likely not be relevant.

If anyone has ever seen a similar question on a GMAT, please let me know!