Intercepts on the Axes made by a Circle

We will learn how to find the intercepts on the axes made by a circle.

The lengths of intercepts made by the circle x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 with X and Y axes are 2$\mathrm{\sqrt{g^{2} - c}}$ and 2$\mathrm{\sqrt{f^{2} - c}}$ respectively.

Proof:

Let the given equation of the circle be x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 ………. (1)

Clearly, the centre of the circle is c (-g, -f) and the radius = $\mathrm{\sqrt{g^{2} + f^{2}- c}}$

Let AB be the intercept made by the given circle on x-axe. Since on x-axis, y = 0. Therefore, x-coordinates of the points A and B are the roots of the equation x$^{2}$ + 2gx + c = 0.

Intercepts on the Axes made by a Circle

Let x$_{1}$ and x$_{2}$ be the x-coordinates of the points A and B respectively. Then, x$_{1}$ and x$_{2}$ also the roots of the equation x$^{2}$ + 2gx + c = 0.

Therefore, x$_{1}$ + x$_{2}$ = - 2g and x$_{1}$x$_{2}$ = c

Clearly the intercept on x-axis = AB

                                          = x$_{2}$ - x$_{1}$ = $\mathrm{\sqrt{(x_{2} - x_{1})^{2}}}$

                                          = $\mathrm{\sqrt{(x_{2} + x_{1})^{2} - 4x_{1}x_{2}}}$

                                          = $\mathrm{\sqrt{4g^{2} - 4c}}$

                                          = 2$\mathrm{\sqrt{g^{2} - c}}$

Therefore, the intercept made by the circle (1) on the x-axis = 2$\mathrm{\sqrt{g^{2} - c}}$

Again,

Let DE be the intercept made by the given circle on y-axe. Since on y-axis, x = 0. Therefore, y-coordinates of the points D and E are the roots of the equation y$^{2}$ + 2fy + c = 0.

Let y$_{1}$ and y$_{2}$ be the x-coordinates of the points D and E respectively. Then, y$_{1}$ and y$_{2}$ also the roots of the equation y$^{2}$ + 2fy + c = 0

Therefore, y$_{1}$ + y$_{2}$ = - 2f and y$_{1}$y$_{2}$ = c

Clearly the intercept on y-axis = DE

                                          = y$_{2}$ - y$_{1}$ = $\mathrm{\sqrt{(y_{2} - y_{1})^{2}}}$

                                          = $\mathrm{\sqrt{(y_{2} + y_{1})^{2} – 4y_{1}y_{2}}}$

                                          = $\mathrm{\sqrt{4f^{2} - 4c}}$

                                          = 2$\mathrm{\sqrt{f^{2} - c}}$

Therefore, the intercept made by the circle (1) on the y-axis = 2$\mathrm{\sqrt{f^{2} - c}}$

Solved examples to find the intercepts made by a given circle on the co-ordinate axes:

1. Find the length of the x-intercept and y-intercept made by the circle x$^{2}$ + y$^{2}$ - 4x -6y - 5 = 0 with the co-ordinate axes.

Solution:

Given equation of the circle is x$^{2}$ + y$^{2}$ - 4x -6y - 5 = 0.

Now comparing the given equation with the general equation of the circle x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0, we get g = -2 and f = -3 and c = -5

Therefore, length of the x-intercept = 2$\mathrm{\sqrt{g^{2} - c}}$ = 2$\mathrm{\sqrt{4 - (-5) }}$ = 2√9 = 6.

The length of the y-intercept = 2$\mathrm{\sqrt{f^{2} - c}}$ = 2$\mathrm{\sqrt{9 - (-5) }}$ = 2√14.

2. Find the equation of a circle which touches the y-axis at a distance -3 from the origin and cuts an intercept of 8 units with the positive direction of x-axis.

Solution:

Let the equation of the circle be x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 …………….. (i)

According to the problem, the equation (i) touches the y-axis

Therefore, c = f$^{2}$ ………………… (ii)

Again, the point (0, -3) lies on the circle (i).

Therefore, putting the value of x = 0 and y = -3 in (i) we get,

9 - 6f + c = 0 …………………… (iii)

From (ii) and (iii), we get 9 - 6f + f$^{2}$ = 0 ⇒ (f - 3)$^{2}$ = 0 ⇒ f - 3 = 0 ⇒ f = 3

Now putting f = 3 in (i) we get, c = 9

Again, according to the problem the equation of the circle (i) cuts an intercept of 8 units with the positive direction of x-axis.

Therefore,

2$\mathrm{\sqrt{g^{2} - c}}$ = 8

⇒ 2$\mathrm{\sqrt{g^{2} - 9}}$ = 8

⇒ $\mathrm{\sqrt{g^{2} - 9}}$ = 4

⇒ g$^{2}$ - 9 = 16, [Squaring both sides]

⇒ g$^{2}$ = 16 + 9

⇒ g$^{2}$ = 25

⇒ g = ±5.

Hence, the required equation of the circle is x^2 + y^2 ± 10x + 6y + 9 = 0.

● The Circle

• Definition of Circle
• Equation of a Circle
• General Form of the Equation of a Circle
• General Equation of Second Degree Represents a Circle
• Centre of the Circle Coincides with the Origin
• Circle Passes through the Origin
• Circle Touches x-axis
• Circle Touches y-axis
• Circle Touches both x-axis and y-axis
• Centre of the Circle on x-axis
• Centre of the Circle on y-axis
• Circle Passes through the Origin and Centre Lies on x-axis
• Circle Passes through the Origin and Centre Lies on y-axis
• Equation of a Circle when Line Segment Joining Two Given Points is a Diameter
• Equations of Concentric Circles
  
• Circle Passing Through Three Given Points
• Circle Through the Intersection of Two Circles
• Equation of the Common Chord of Two Circles
• Position of a Point with Respect to a Circle
• Intercepts on the Axes made by a Circle
• Circle Formulae
• Problems on Circle

11 and 12 Grade Math 

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