# circle geomtery help

• Jun 1st 2009, 04:44 AM
Tweety
circle geomtery help
The circle C has centre (5, 2) and passes through the point (7, 3).

(a) Find the length of the diameter of C.
$\displaystyle 2x \sqrt{4+1} = 2\sqrt{5}$

(b) Find an equation for C.
$\displaystyle (x-5)^2 + (y-2)^2 = 5$

(c) Show that the line y = 2x − 3 is a tangent to C and find the coordinates
of the point of contact.

Need help with question 'c'.

I know that the angle between a tangent and radius is 90 , but how would I show that the product of their gradients is -1, if I dont know the point of contact?

Also how do I find the point were it touches the circle?
• Jun 1st 2009, 05:14 AM
mr fantastic
Quote:

Originally Posted by Tweety
The circle C has centre (5, 2) and passes through the point (7, 3).

(a) Find the length of the diameter of C.
$\displaystyle 2x \sqrt{4+1} = 2\sqrt{5}$

(b) Find an equation for C.
$\displaystyle (x-5)^2 + (y-2)^2 = 5$

(c) Show that the line y = 2x − 3 is a tangent to C and find the coordinates
of the point of contact.

Need help with question 'c'.

I know that the angle between a tangent and radius is 90 , but how would I show that the product of their gradients is -1, if I dont know the point of contact?

Also how do I find the point were it touches the circle?

Here's one approach out of many possible approaches: Show that the two equations

$\displaystyle y = 2x - 3$

$\displaystyle (x-5)^2 + (y-2)^2 = 5$

have only one solution.
• Jun 1st 2009, 05:23 AM
Tweety
Quote:

Originally Posted by mr fantastic
Here's one approach out of many possible approaches: Show that the two equations

$\displaystyle y = 2x - 3$

$\displaystyle (x-5)^2 + (y-2)^2 = 5$

have only one solution.

So if I solved this simutaneously, wouldn't that mean that these to equations intersect at that one point, instead of touching?
• Jun 1st 2009, 05:24 AM
mr fantastic
Quote:

Originally Posted by Tweety
So if I solved this simutaneously, wouldn't that mean that these to equations intersect at that one point, instead of touching?

If a line intersects a circle at one point then it is tangent to the circle at that point.