Question

determine the equation of the circle passing through the point(3,5) and having its center at the intersection of the lines x-y =6 and 2x +3y+8=0​

Answer 1

Given :

The circle passing through point = 3 , 5

The center at the intersection lines are x - y = 6  and  2 x + 3 y = - 8

Find :

The equation of circle

Solution :

∵  The lines are  x - y = 6  and  2 x + 3 y = - 8

Solving lines equation

x - y = 6                   .........1  

2 x + 3 y = - 8           ..........2

i.e

 ( 2 x + 3 y ) + 3 ( x - y ) = - 8 + 3 × 6

Or, ( 2 x + 3 x ) + ( 3 y - 3 y ) = - 8 + 18

Or,  5 x + 0 = 10

Or,  5 x = 10

∴        x = $\dfrac{10}{5}$ = 2

Put the value of x into eq 1

 2 - y = 6

Or,  y = 2 - 6 = - 4

∵ The circle passes through points $x_1 , y_1$ = 3 , 5

                                                          $x_2 , y_2$ = 2 , - 4

Distance between points  $x_1 , y_1$ ,  $x_2 , y_2$  = r = $\sqrt{( x_2-x_2)^{2}+(y_2-y_1)^{2} }$

                                                                       = $\sqrt{( 2-3)^{2}+(-4-5)^{2} }$

                                                                       = $\sqrt{82}$

∵ The standard equation of circle = $(X - x_1 )^{2}$ + $(Y - y_1)^{2}$ = r²

 where r = radius

And r =  $\sqrt{82}$

Or,   $(X - 3 )^{2}$ + $(Y - 5)^{2}$ = ( $\sqrt{82}$ )²

∴      $(X - 3 )^{2}$ + $(Y - 5)^{2}$ = 82

Hence, The equation of circle is     $(X - 3 )^{2}$ + $(Y - 5)^{2}$ = 82   Answer