A circle C has centre (-5, 12) and passes through the point (0,0) Find the second point where the line y=x intersects the circle.
The equation of a circle comes in a standard format of (x-a)2 + (y-b)2 = r2 where (a, b) are the coordinates of the centre of the circle and r is the radiusFrom the information, we can find the radius of the circle (a diagram is useful) by finding the distance between the centre point and a point on the circle. Hence we can use the distance between (-5,12) and (0,0).Using Pythagoreas, the distance is ((x-x)2 + (y-y)2)1/2 Hence the radius2 is (0--5)2 + (0-12)2 i.e. r2 = 169 and r = 13Hence the equation of the circle is (x+5)2 + (y-12)2 = 169
The next part would involve finding the other intersection with the circle. Therefore we need the solutions to the simultaneous equations y= x and (x+5)2 + (y-12)2 = 169. Achieve this by substituting substituting in x for y (as y=x)Hence (x+5)2+(x-12)2=169Expanding this: x2 + 10x + 25 + x2 -24x + 144 = 1692x2-14x+169 = 169So 2x2-14x = 0x2-7x = 0Factorise thisx(x-7) = 0Hence x = 0 or x = 7Using y = x, the points of intersection are (0,0) and (7,7).(0,0) is given so the other point is (7,7).
