I'm Cikadamate, I joined just because I am a year ahead in maths but I'm struggling. I am hoping that when necessary I will be able to use this forum to help me understand anything I am stuck on.
Right now we're doing co-ordinate geometry and This is the question which prompted me to join:
A circle has centre G(2,-1). A point H(a,3) on the circumference of the circle is joined to the centre of the circle. if the radius of the circle is 8, find 'a'.
What I need to know is the method for working this type of question out.
Any help would be most appreciated
i just drew a circle centered at (2,-1) of radius 8. we want to know the x-coordinate(s) of the point(s) where this circle intersects the line y = 3.
if one draws a right triangle, formed by (2,-1), (a,-1) and (a,3), one sees immediately that the hypotenuse has length 8, the vertical leg has height 4 ( = 3 - (-1)), and the horizontal leg has length |a-2| (a could be to the left of the center).
by the pythagorean theorem:
(a-2)^{2} + 16 = 64, just as HallsofIvy derived.
(EDIT: there are good reasons why all three answers given to you will lead to the same answer. in a sense, circles ARE "distance measuring curves", that is: the formula for distance, and the formula for the equation of a circle are two different ways to interpret the pythagorean equation:
A^{2} + B^{2} = C^{2}.
circles hold "C" constant, and let "A" and "B" vary, while distance interprets "C" as a function of "A" and "B". in other words, circles implicitly define one leg length of the inscribed right triangles as a function of the other leg's length, while the distance formula explicitly calculates the hypotenuse, given the legs.
i often find it amazing that "pointy" things like triangles, and "line length" things (like distance) are so intimately bound up with the notion of a circle, which is neither pointy nor "linear". this "pythagorean connection" often shows up in unexpected areas in many branches of mathematics. it's THAT deep.)