Question

точка В семестрична в точці А відносно О. Знайти координати точки В, якщо А(11;-7) О(5;3). Скласти рівняння кола з центром в точці А радіуса АВ, пожалуйста срочно помогите, прошу, 40 балов даю

Answer 1

Ответ:    B(- 1 ; 13 ) ;   ( x - 11 )² + ( y + 7 )² = 544 .

Пошаговое объяснение:

А( 11 ; - 7 ) , О( 5 ; 3 ) . Нехай коорд. т. В( х₂ ; у ) . Точка О - середина

відрізка АВ , тому

( 11 + х₂ )/2 = 5 ; ----> 11 + х₂ = 10 ; ----> x₂ = 10 - 11 ; ---->  x₂ = - 1 ;

( - 7 + y₂)/2 = 3; ---->  - 7 + y₂= 6 ; ----> y₂ = 6 + 7 ;  ---->  y₂ = 13 .  B(- 1 ; 13 ).  

AB = r = √[ (- 1 - 11 )² + (13 + 7 )² ] = √( 144 + 400 ) = √544 = 4√34 .

Рівняння кола знайдемо у вигляді :   ( x - a )² + ( y - b )² = r ² :

 ( x - 11 )² + ( y + 7 )² = ( √544 )² ;

  ( x - 11 )² + ( y + 7 )² = 544 .

Answer 2

Finding the coordinates of point B

To find the coordinates of point B, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) are given by ((x1 + x2)/2, (y1 + y2)/2).

Given that point A has coordinates (11, -7) and point O has coordinates (5, 3), we can find the coordinates of point B as follows:

x-coordinate of B = (x-coordinate of A + x-coordinate of O)/2 = (11 + 5)/2 = 16/2 = 8 y-coordinate of B = (y-coordinate of A + y-coordinate of O)/2 = (-7 + 3)/2 = -4/2 = -2

Therefore, the coordinates of point B are (8, -2).

Equation of the circle with center A and radius AB

To find the equation of the circle with center A and radius AB, we can use the standard form of the equation of a circle. The standard form of the equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.

In this case, the center of the circle is A(11, -7) and the radius is the distance between A and B, which is AB. We can use the distance formula to find the length of AB.

The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by √((x2 - x1)^2 + (y2 - y1)^2).

Using the coordinates of A(11, -7) and B(8, -2), we can calculate the length of AB as follows:

AB = √((8 - 11)^2 + (-2 - (-7))^2) = √((-3)^2 + 5^2) = √(9 + 25) = √34

Therefore, the equation of the circle with center A and radius AB is (x - 11)^2 + (y + 7)^2 = 34.

I hope this helps! Let me know if you have any further questions.