Question

Points S(6, −2a) and T(4b, 6) lie in the circumference with the centre (7, −2a). Line segment ST is a diameter of the circle. What are the values of a and b ?

Answer 1

Точки S(6, −2a) и T(4b, 6) лежат на

окружности с центром (7, −2a). Отрезок прямой ST-это диаметр окружности. Каковы значения a и b ?

Объяснение: Центр окружности, для диаметра, является серединой.

По формулам середины отрезка

7=(6+4b) :2, 14=6+4 b, b=2;

-2a=(-2 a+6) :2 , -4 a=-2a+6 , a=-3.

Answer 2

Since ST is a diameter of the circle with center (7, -2a), the midpoint of ST should be equal to the center of the circle.

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

So, for ST, the midpoint is:

Midpoint of ST = ((6 + 4b) / 2, (-2a + 6) / 2)

Since ST is a diameter, the midpoint should be equal to the center of the circle, which is (7, -2a):

((6 + 4b) / 2, (-2a + 6) / 2) = (7, -2a)

Now, let's solve for a and b.

1. Equate the x-coordinates: (6 + 4b) / 2 = 7

Multiply both sides by 2: 6 + 4b = 14

Subtract 6 from both sides: 4b = 14 - 6 4b = 8

Divide by 4: b = 8 / 4 b = 2

2. Equate the y-coordinates: (-2a + 6) / 2 = -2a

Multiply both sides by 2: -2a + 6 = -4a

Add 2a to both sides: 6 = -4a + 2a

Combine like terms: 6 = -2a

Now, divide by -2 to find the value of a: a = 6 / -2 a = -3

So, the values of a and b are: a = -3 b = 2