Answer the following :

Find the lengths of the intercepts made on the co-ordinate axes, by the circle:

x^{2} + y^{2} – 5x + 13y – 14 = 0

#### Solution

To find x-intercept made by the circle x^{2} + y^{2} + 2gx + 2fy + c = 0, substitute y = 0 and get a quadratic equation in x, whose roots are, say, x_{1} and x_{2}

These values represent the abscissae of ends A and B of x – intercept.

Length of x – intercept = | AB | = | x_{2} – x_{1} | Similarly, substituting x = 0, we get a quadratic equation in y whose roots, say, y_{1} and y_{2} are ordinates of the ends C and D of y-intercept. Length of y – intercept = | CD | = | y_{2} – y_{1} |

Given equation of the circle is

x^{2} + y^{2} – 5x + 13y – 14 = 0 …(i)

Substituting y = 0 in (i), we get

x^{2} – 5x – 14 = 0 …(ii)

Let AB represent the x-intercept, where

A = (x_{1}, 0), B = (x_{2}, 0).

Then from (ii),

x_{1} + x_{2} = 5 and x_{1}x_{2} = – 14

(x_{1} – x_{2})^{2} = (x1 + x2) 2 – 4 x1x2

= (5)^{2} – 4(– 14)

= 25 + 56

= 81

∴ | x_{1} – x_{2} | = `sqrt((x_1 - x_2)^2) = sqrt(81)` = 9

∴ Length of x-intercept = 9 units

Substituting x = 0 in (i), we get

y^{2} + 13y – 14 = 0 …(iii)

Let CD represent the y-intercept, where

C = (0, y_{1}), D = (0, y_{2}).

Then from (iii),

y_{1} + y_{2} = – 13 and y_{1} y_{2} = – 14

(y_{1} – y_{2})^{2} = (y1 + y2)^{2} – 4 y_{1} y_{2}

= (– 13)^{2} – 4(– 14)

= 169 + 56

= 225

∴ | y_{1} – y_{2} | = `sqrt((y_1 - y_2)^2) = sqrt(225)` = 15

∴ Length of y-intercept = 15 units