Interesting geometry/coordinate plain question

[adilka]

I found this one to be interesting. Took me 8 mins to solve and still had to semi-guess (thankfully correctly). A shorter solution is a clever one.

What is the least possible distance between a point on the circle X^2+y^2 = 1 and a point on the line y=0.75x-3

A) 1.4
B) Sqrt(2)
c) 1.7
d) Sqrt(3)
e) 2.0


OA: A

[logitech]

I found this one to be interesting. Took me 8 mins to solve and still had to semi-guess (thankfully correctly). A shorter solution is a clever one.

What is the least possible distance between a point on the circle X^2+y^2 = 1 and a point on the line y=0.75x-3

A) 1.4
B) Sqrt(2)
c) 1.7
d) Sqrt(3)
e) 2.0


OA: A

X^2+y^2 = 1 is a circle with radius 1 around the center of X,Y plane

y=0.75x-3 for x=0, y =-3 and for y=0 x =4 So it forms 3,4,5 right triangle.

The shortest line should be perpendicular to the line , that goes to the center of the circle , so it is actually the HEIGHT of the triangle

The area of the triangle can be written in two forms to find this height

(3x4)/2 = (5xh)/2 so this gives us 12/5

well 1 is the radius , so the distance is 12/5 - 1 = 7/5 = 1.4.

It is long to explain but I solved this under a minute.

[acecoolan]

.

It is long to explain but I solved this under a minute.

Thats why ..u r the boss !!!!

Thanks logitech

[adilka]

Great solution. That's exactly how it was explained. I failed to make a connection between the height and the area, so was solving for the height in a different way.
Great job, logitech