I did do a graph drawing, but from my drawing it looks to me that the closest point the origin isn't the straight line that leads upwards to 5, but instead the continuation of the line that passes through the origin and the center of the circle.

To do an accurate diagram, I would need to use a compass. :P Is using a compass something you would recommend?

If a circle has radius R, center C, and the distance of C from the origin is r, then the closest point between circle and origin is a distance |R-r| from the origin. You should not be worrying about lines at all.

Reply 9

DFranklin

Original post by UCASLord

The point on the circle x^2 + y^2 + 6x + 8y = 75 which is closest to the origin, is how far from the origin.

My solution:

(x + 3)^2 + (y + 4)^2 = 100

Line from origin to circle can be constructed using the coordinates of the center of the circle and the origin.

So y = 3/4x is the line.

(x + 3)^2 + (3/4x + 4)^2 = 100

x = 3.6621I'm assuming you used a calculator for this. It should be stating the obvious, but if you need to use a calculator in the MAT, you've gone wrong somewhere

when x = 3.6624, then y = 4.102

Note that for your values of x and y, x^2 = 6.6624^2 > 36, (y+4)^2 = 8.012^2 > 64, so (x+3)^2+(y+4)^2 is definitely *not* 100.

It is usually easier to check solutions that to find them in the first place, and so checking is usually a good thing to do...

Reply 10

DFranklin

Original post by UCASLord

Thanks for your response, but I'm afraid that y = 4/3x is what I got for the line equation the first time, that's a typo in the above post.

What I do next is I insert 4/3x into the circle equation, like so:

(x + 3)^2 + (4/3x + 4)^2 = 100

This leads me to the Quadratic 25/9 (x^2) + 26/3 (x) - 75...
(x+3)^2 = x^2 + 6x + 9
(4x/3 + 4)^2 = 16x^2 / 9 + 32x/3 + 16 (***)

so we end up with 25/9 x^2 + 50/3 x - 75 = 0.

I'm guessing you somehow lost a factor of 4 in the 'x' term when squaring to find (***).

(edited 6 years ago)

Reply 11

UCASLord

-RM-

(edited 6 years ago)

Reply 12

DFranklin

Original post by UCASLord

Hmm, that seems to give the distance of 3 (because 10 - 3 - 4 = 3), but the actual distance is 5.

The circle has center (-3, -4) which is distance 5 from the origin. It has radius 10. The distance given by |R-r| is therefore |10 - 5| = 5.

Edit: did you think the distance of (-3, -4) from the origin was 7?!?