Find the equation of a circle if the circle is tangent to the line -x y40 and the center is on the line x 2y30?

Answer 1

The equation of a circle is equal to: (x - a)2 + (y - b)2 = r2 Where (a, b) is equal to the center of the circle and r = radius. To understand more why this is the equation of a circle, think of a circle with a known center point (a,b). A circle, by definition, is the set of all points an equal distance from a given point (the center). That distance is also known as the radius. So if only we had some formula to calculate the radius, we would have our equation. Luckily we do. In your circle, imagine a right triangle with the hypotenuse being the radius (wikipedia has a good picture of this at http://en.wikipedia.org/wiki/Circle). Using the pythagorean theorem we have a formula for calculating the radius. Remember that the theorem states that: a2 + b2 = c2 Where a and b are the lengths of the two sides of the right triange adjacent to the right angle and c is the hypotenuse. For a circle with a center point at the origin, the length of the two sides a and b are simply x and y; howver if the center is not at the origin the length of each adjacent side is (x-a) and (y-b) (again draw the circle and triangle out or refer to the wikipedia drawing to better see this). Since c = radius, we now can substitute in and get: (x - a)2 + (y - b)2 = r2 Which we know is the equation of a circle In this problem, we don't know the exact center point or radius yet, we only know the equations of lines that intersect the center and are tangent to the circle. From my memory, this problem needs at least one more piece of information (the radius) before it can be solved. If someone else knows how to do this without the radius, please have at it; otherwise (to the poster) double check to make sure the problem you have did not specify a radius.