Hello everyone!

Find a point P on the circle such that is minimum, where is the origin and is the x-axis.

I tried sketching the curve but that didn't help. Parametric coordinates are the key? Or any other approach?

Printable View

- Apr 23rd 2010, 12:59 AMfardeen_genFind point on circle such that angle is minimum?
Hello everyone!

Find a point P on the circle such that is minimum, where is the origin and is the x-axis.

I tried sketching the curve but that didn't help. Parametric coordinates are the key? Or any other approach? - Apr 23rd 2010, 02:27 AMArchie Meade
Hi fardeen_gen,

you can locate the circle by finding it's centre and radius.

Then it's possible to work geometrically from that.

complete the squares for x and y

hence the radius is 2, centre is (2,3)

Referring to the attachment, there are identical triangles

with perpendicular sides of lengths 2 and 3.

Hence

Hence the angle at the origin in the triangle on the x-axis is

We can now find the x and y co-ordinates of P.

- Apr 23rd 2010, 06:13 AMfardeen_gen
I hadn't thought about making a line passing through the centre. Very nice :)

Other users are welcome to solve it by alternate methods. - Apr 23rd 2010, 10:22 AMArchie Meade
Here's another way,

if we differentiate the circle equation,

we will have a function describing the tangent slope

at all points (x,y) on the circle.

this is the tangent slope at any point (x,y) on the circle circumference.

We are looking for the line of the form y=mx that passes through

the origin, since y=mx+c has c=0.

This is the tangent that goes through the origin.

Hence

If we rearrange this we can place the x and y into the circle equation.

hence we can replace the in the circle equation

obtaining a linear equation in x and y at the point of tangency.

Therefore

This line goes through the point of tangency but crosses the y-axis at y=3.

This is because there are 2 tangents to the circle that pass through the origin..

the second being the y-axis.

Hence our resulting linear equation is the line passing through both points of tangency

for lines that go through the origin.

We want the line that goes through this point and the origin.

Hence

The x co-ordinate is 2.77

Hence we can find the y co-ordinate and write the tangent slope and equation.