Maharashtra State BoardHSC Science (General) 11th
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Answer the following : Find the lengths of the intercepts made on the co-ordinate axes, by the circle: x2 + y2 – 5x + 13y – 14 = 0 - Mathematics and Statistics

Sum

Answer the following :

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

x2 + y2 – 5x + 13y – 14 = 0

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Solution

To find x-intercept made by the circle x2 + y2 + 2gx + 2fy + c = 0, substitute y = 0 and get a quadratic equation in x, whose roots are, say, x1 and x2

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

Length of x – intercept = | AB | = | x2 – x1 | Similarly, substituting x = 0, we get a quadratic equation in y whose roots, say, y1 and y2 are ordinates of the ends C and D of y-intercept. Length of y – intercept = | CD | = | y2 – y1 |

Given equation of the circle is

x2 + y2 – 5x + 13y – 14 = 0 …(i)

Substituting y = 0 in (i), we get

x2 – 5x – 14 = 0 …(ii)

Let AB represent the x-intercept, where

A = (x1, 0), B = (x2, 0).

Then from (ii),

x1 + x2 = 5 and x1x2 = – 14

(x1 – x2)2 = (x1 + x2) 2 – 4 x1x2

= (5)2 – 4(– 14)

= 25 + 56

= 81

∴  | x1 – x2 | = `sqrt((x_1 - x_2)^2) = sqrt(81)` = 9

∴ Length of x-intercept = 9 units

Substituting x = 0 in (i), we get

y2 + 13y – 14 = 0 …(iii)

Let CD represent the y-intercept, where

C = (0, y1), D = (0, y2).

Then from (iii),

y1 + y2 = – 13 and y1 y2 = – 14

(y1 – y2)2 = (y1 + y2)2 – 4 y1 y2

= (– 13)2 – 4(– 14)

= 169 + 56

= 225

∴ | y1 – y2 | = `sqrt((y_1 - y_2)^2) = sqrt(225)` = 15

∴ Length of y-intercept = 15 units

Concept: Equation of a Circle
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 6 Circle
Miscellaneous Exercise 6 | Q II. (11) (ii) | Page 137
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