Circles

Question 1

What is the general equation of a circle?

Answer 1

(x - a)² + (y - b)² = r²
Centre of circle is (a,b), radius of circle is r.

Question 2

Upon expanding the general equation of a circle, what are we left with?

Hint: values such as g, f, and c exist.

Answer 2

x² + y² + 2gx + 2fy + c = 0

g = -a, f = -b, c = a² + b² - r²

Centre of circle is (a, b)

Question 3

Solving the expansion of the general equation of the circle using completing the square method, what do we finally have?

Answer 3

(X + g)² + (y + f)² = g² + f² - c

Question 4

From the final equation of the circle [ (x + g)² + (y + f)² = g² + f² - c ] how can the radius of said circle be found?

Answer 4

r = √ (g² + f² - c)

Question 5

What is the length of a tangent from point P(X,Y) outside a circle?

Answer 5

L = (X² + Y² + 2gX + 2fY + c)^½

Question 6

What is the equation of a tangent at point (x₁, y₁) on a circle?

Answer 6

xx₁ + yy₁ + g(x + x₁) +f(y +y₁) + c = 0