Intercept made by the circle on the axis:

Assume that the equation of circle be x² + y² + 2gx + 2fy + c = 0.........(1)

X- INTERCEPT: Intercept made by circle on x-axis is called as X-intercept. The circle will intersect x-axis where y = 0 => x² + 2gx + c = 0       ...(2)

The three cases arises here

Case I:     If discriminant > 0 that is g²-c >0  circle will intersect the axis at 2 distinct and real points let A (x₁, 0) and B (x₂, 0). Length of intercept

                                    .  

Case II:    If discriminant = 0 that is g²=c. Then the circle will touch the x-axis. In this particular case length of the intercept made by the circle on x-axis will be zero.

Case III: If discriminant < 0 that is g²-c <0, in this case circle will neither touch nor intersect x-axis

Y- INTERCEPT: Intercept made by circle on y-axis is called as Y-intercept. The circle will intersect y-axis where x = 0=>y² + 2fy + c = 0        ...(2)

Again the three cases arises

Case I:     When discriminant > 0 i.e. f²-c >0 circle will intersect y-axis at 2 distinct and real points say A(0, y₁) and B(0, y₂). Length of y-intercept

Case II:    If discriminant = 0 i.e. f² - c=0 Then circle will touch the axis. In this case length of intercept made by the circle on x-axis will be zero.

Case III: If discriminant < 0, that is f²-c <0 in this case circle will neither touch nor intersect axes.

Illustration: Find the equation of circle touching y-axis at (0, 3) and making intercept of 8 units on x-axis.

Solution: Let equation of circle be x² + y² + 2gx + 2fy + c = 0

             putting x = 0, we get y² + 2fy + c = 0                     ... (1)

             As it touches y-axis at (0, 3), (1) should be of the form (y - 3)² = 0

             => y² - 6x + 9 = 0

             Comparing, we get, f = - 3 and c = 9

              Putting y = 0, we get x² + 2gx + c = 0

              So, 

               => g² - c = 16 => g² = 25 => g = ± 5

               so, equation is x² + y² ± 10x - 6y + 9 = 0

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