Question

➜Find the equation of a circle touching the X-axis and equation diameters are x - y = 1 1and 2x + y = 5 .

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Answer 1

x - y = 1

2x + y = 5

+

3x + 0 = 6

3x = 6

x = 6/3

x = 2

therefore, 2x + y = 5

2 * 2 + y = 5

4 + y = 5

y = 5 - 4

y = 1

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Answer 2

Answer:

Equation of diameter of circle are-

$x - y = 1 \: \: - - - (1)$

and

$2x + y = 5 - - - (2)$

Adding equation (1) and (2)

$3x = 6 \\ x \frac{6}{3} \\ x = 2$

put x=2 in equation (1)

$x - y = 1 \\ 2 - y = 1 \\ - y = 1 - 2 \\ - y = - 2 \\ y = 2$

coordinates of circles ,O=(2,1)

Now, further given that circle touches the x - axis. So, as we know that radius and x - axis are perpendicular to each other. So, radius of circle, r = Distance of center from x - axis = 1 unit.

Required equation of circle having center (2, 1) and radius, r = 1 is given by,

${x}^{2} + 4 - 4x + {y}^{2} + 1 - 2y = 1 \\ {x}^{2} + 4 - 4x + {y}^{2} - 2y = 1 - 1 \\ {x}^{2} + 4 - 4x + {y}^{2} - 2y = 0 \\ {x}^{2} - {y}^{2} - 4x - 2y + 4 = 0$

Hence,Required equation of circle is

${x}^{2} + {y}^{2} - 4x - 2y + 4 = 0$