Question

Heya everyone ,

Find the equation of the circle passing through the points (2,-1) , (2,3) and (4,-1)???

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Answer 1

Hope u like my process
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Let Equation of circle be
x² +y² +Dx +Ey +F=0____(1)
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(2,-1) : 2² + (-1)² +D×2 +E×(-1) +F =0

= > 4 +1 +2D - E + F =0

=> F = E - 2D - 5 ___(2)
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(2,3): 2² +3² + D×2 +E×3 +F = 0

=> 4 + 9 + 2D +3E + F = 0

Substituting the value of F from (2)

=> 2D +3E +E-2D - 5 +13=0

=> 4E = - 8

=> E = - 2
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(4,-1) : 4² + (-1)² +D×4 +E×(-1) +F =0

=> 16 + 1 + 4D - E +F = 0

=> 17 +4D - E +E - 2D - 5 =0

=> 2D = 5 - 17 =-12

=> D = - 6
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Putting the value of D and E in eq(2)
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=> F = E - 2D - 5

or, F = - 2 - 2(-6) - 5

or, F = - 2 +12 - 5= 5

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Putting all values in eq(1)
++++++++++++++++++++++

Thus the required equation of circle is

=> x² + y² +(-6)x +(-2)y + 5 = 0

=> x² + y² - 6x - 2y +5 =0 ___(answer)

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Hope this is ur required answer

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Answer 2

Solution....

Let the equation of the circle be ----

${x}^{2} + {y}^{2} + 2gx + 2fy + c = 0...................... \: \: eq.(1.)$

Now, the circle passes through the points [2,-1]. So, we put x = 2 and y = -1 in eq. (1.)

${2}^{2} + {( - 1)}^{2} + 2 \times g \times 2 + 2 \times f \times( - 1) + c = 0$

4 + 1 + 4g - 2f + c = 0

5 + 4g -2f + c = 0

4g - 2f + c = - 5 ................. Eq.[2.]

Now, the circle with eq.[1.] passes through the points [ 2,3 ]. So, put x = 2 and y = 3 in eq (1.).

${2}^{2} + {3}^{2} + 2 \times g \times 2 + 2 \times f \times 3 + c = 0 \\ \\ 4 + 9 + 4g + 6f + c = 0 \\ \\ 4g + 6f + c = - 13.................. \: \: eq(3.)$

Now, the circle with eq.(1) passes through the points [4,-1] . So, we put x = 4 and y = 1 in eq.(1)

${4}^{2} + {-1}^{2} + 2 \times g \times 4 + 2 \times f \times -1 + c = 0 \\ \\ 16 + 1 + 8g - 2f + c = 0 \\ \\ 8g - 2f + c = - 17................. \: \: eq.(4.)$

Now Subtract eq. 3 from eq. 2
we get,

- 8f = 8

▪▪f = -1 ▪▪

Now, multiply eq.(3) by 2 and then subtract from eq.4

we get,

-14f - c = 9

But f = - 1

- 14 × -1 - c = 9

▪▪c = 5 ▪▪

Now put value of f & c in eq.2

4 × g - 2 ×(-1 ) + 5 = -5

4g + 2 = -5 -5

4g = -10 -2

4g = -12

g = -12 / 4

▪▪ g = -3 ▪▪

Finally put values of g , f & c in eq. [1.]

${x}^{2} + {y}^{2} + 2 \times - 3 \times x \: + 2 \times ( - 1) \times y + ( 5 ) = 0 \\ \\ {x }^{2} + {y}^{2} - 6x - 2y + 5 = 0........... \: is \: the \: required \: equation \: of \: circle.$