M13-12

Question

What is the smallest possible distance between a point on the circle defined by the equation  x^2 + y^2 = 1  and a point on the line described by the equation  y = \frac{3}{4} x - 3 ?

A.  1.4  
B.  \sqrt{2}  
C.  1.7  
D.  \sqrt{3}  
E.  2.0

Bunuel · Accepted Answer

Official Solution:

What is the smallest possible distance between a point on the circle defined by the equation  x^2 + y^2 = 1  and a point on the line described by the equation  y = \frac{3}{4} x - 3 ?

A.  1.4  
B.  \sqrt{2}  
C.  1.7  
D.  \sqrt{3}  
E.  2.0

Check the diagram below:
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/m13-12.png 
 
Observe that the circle represented by the equation  x^2+y^2 = 1  is centered at the origin and has a radius of  r= \sqrt{1}=1 .
 
The minimum distance from the circle to the line would be  the length of the perpendicular from the origin to the line (since the circle is centered at the origin) minus the radius of the circle (which is 1) .
 
Now, we must find the length of the perpendicular, or the height of the right triangle formed by the  X  and  Y  axis and the line  y = \frac{3}{4}x-3 .
 
The legs of this triangle have lengths  leg_1=4  and  leg_2=3 .
 
Therefore, we have a right triangle with legs 4 and 3 and a hypotenuse of length 5. What is the height of this triangle (the perpendicular from the right angle to the hypotenuse)? Since the perpendicular to the hypotenuse always divides the triangle into two triangles with the same properties as the original triangle, we can use the following proportion:  \frac{height}{leg_1}=\frac{leg_2}{hypotenuse} . Thus,  \frac{height}{3}=\frac{4}{5} , which results in a height of  2.4 .
 
The minimum distance between the circle and the line is:  distance=height-radius=2.4-1=1.4 .

Answer: A

him1985 · Answer

I am not comfortable with graphical solution. Is there any way to solve such question without graph.

Bunuel · Answer

Check here: what-is-the-least-possible-distance-between-a-point-on-the-85184.html

Mrtinhnv · Answer

Other solution:
Area of right triangle (3-4-5): 1/2*3*4 = 6
Since (r+x is height of triangle) => Area = 1/2*(1+x) = 6 
 => 5x = 7 => x=1.4
Is it right>

Monocle · Answer

I like it Mrtinhnv.

Another way of finding height of triangle:

Area= 1/2*b*h
=1/2*4*3 = 6

therefore: 
6=1/2*5*x
12=5*x
x=12/5

So 12/5-1 = 1.4

PerseveranceWins · Answer

I think this is a  high-quality  question.

Senthil7 · Answer

I think this is a  high-quality  question and I agree with explanation.

iamanorwhal · Answer

After plotting the equations out, I solved it by calculating the Area to find the height (from Origin to the Hypotenuse)

Step 1: Draw the circle and the line. You will immediately notice that the triangle is a 3-4-5 right triangle.
Step 2: Calculate the Area = (3*4)/(2) = 6
Step 3: Calculate the height from the origin to the hypotenuse by using the Area calculated earlier and the hypotenuse as the Base

(Base * Height)/(2) = 6
(5 * H)/(2) = 6
(5H) = 12
(H) = (12/5) = 2.4

***Height = 2.4

Step 4: Subtract the radius of 1 from height of 2.4: (2.4 - 1.0) =  1.4 (answer choice A)

Shiridip · Answer

If we use the property  median from right angle is half of the hypotenuse  then the height = 5/2 = 2.5. so the answer is 2.5-1 = 1.5? What am I missing?

pranjal123 · Answer

Hi Shiridip The Shortest distance is the perpendicular distance.Median connects the midpoint of hypotenuse to opposite vertex but is not perpendicular. In case you need a relation between Altitude(h) and sides: h= \frac{ab}{c} , where a and b are the legs measures, and c is the hypotenuse measure. Thanks.

+1 Kudos if this post helps .

mahrah · Answer

I understood the least possible distance between a point   on   the circle to mean any point on the circumference to the line. How do we know to deduce it means from the origin of the circle to the line?

Bunuel · Answer

Yes, it's the distance between the point on the circumference to the line. We just find the distance from the centre and then subtract the length of the radius:

Now, the minimum distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin)  minus the radius of a circle (which is 1) .

Chidinho · Answer

Hi Bunuel, please how was leg1 and leg 2 obtained

Bunuel · Answer

Those are the values of x and y intercepts of y = 3/4*x - 3. To get these values put x = 0 for y-intercept and y = 0 for x-intercept.

Charanbas · Answer

I think this is a  high-quality  question and I agree with explanation.

16SK16 · Answer

I think this is a  high-quality  question and I agree with explanation.

RK007 · Answer

I think this is a  high-quality  question and I agree with explanation.

anupam87 · Answer

Hi  Bunuel ,

Could you please explain how we deduced the x intercept of the line as 4? I can understand that Y intercept is -3.

Thanks

Bunuel · Answer

x-intercept is the value of x when y = 0. Put y = 0 into y = 3x/4 - 3 to get the value of x.

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