Question

Hello guys....pls help me with this problem. ......with clear explanation.
The equation of the circle passing through the points (1,-2) and (4,-3) and whose centre lies on the line 3x+4y=7.

Answer 1

Points = ( 1, - 2) and ( 4, - 3 )

Equation of line = 3x + 4y = 7

Equation of circle = x^2 + y^2 + 2gx + 2fy + c = 0

It passes through points ( 1, - 2 ) and ( 4, - 3 )

1^2 + ( - 2 ) ^2 + 2 g ( 1 ) + 2f ( - 2 ) + c = 0

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

5 + 2g - 4f + c = 0 ---> ( i )

4^2 + ( - 3 ) ^2 + 2g ( 4 ) + 2f ( - 3 ) + c = 0

16 + 9 + 8g - 6f + c = 0

25 + 8g - 6f +c = 0 ---> ( ii )

Subtracting equation ( i ) and ( ii ),

=> 5 + 2g - 4f +c - 25 - 8g + 6f - c = 0

=> - 20 - 6g +2f = 0

=> - 6g + 2f = 20

=> 2 ( - 3g + f) = 20

=> - 3g + f = 20 /2 = 10

=> 0 = 10 + 3g - f ---> ( iii )

From Eqn, 3x + 4y = 7

The centre ( - g, - f ) lies on this Eqn,

So, - 3g - 4f = 7 [ Replacing x to g and y to f ]

-3g - 4f - 7 = 0 ---> ( iv )

Solving ( iii) and ( iv ),

g = - 47 / 15 and f = 3 / 5

Putting these values in Eqn I,

5 + 2 ( - 47 / 15 ) - 4 ( 3 /5 ) + c = 0

5 - 94 / 15 - 12 / 5 + c = 0

( 45 - 94 - 36 ) /15 + c= 0

85 / 15 + c = 0

17 / 3 + c = 0

c = - 17 /3

Putting the values of g, f and c in Eqn of circle,

x^2 + y^2 + 2gx + 2fy + c = 0

x^2 + y^2 + 2 ( - 47 / 15) x + 2 ( 3 /5 )y - 17 /3 = 0

x^2 + y^2 - 94 x /15 + 6 y/ 5 - 17 /3 = 0

$<h4>15x^2 + 15 y^2 - 94x - 18y - 85 = 0$