Answer:
The equation of the circle is x² + y² + 12x - 14y - 165 = 0 , its center is (-6, 7) and its radius is 5√10 units.
Step-by-step explanation:
a. The center of the circle
Since the equation of the tangent of the circle is 3x - y = 15, we write it in standard form. So y = -3x - 15. Its gradient is -3. The perpendicular line tot he tangent is the radius and it passes through the center of the circle and the point (9, 12). For two perpendicular lines of gradient m₁ and m₂, m₁m₂ = -1. If m₁ = -3, then m₂ the gradient of our radius line is gotten from m₁m₂ = -1
m₂ = -1/m₁ = -1/-3 = 1/3
Since the radius also passes through the point (9, 12), its equation is gotten from
(y - y₁)/(x - x₁) = m₂ where (x₁, y₁) = (9, 12)
Substituting the value of the variables into the equation, we have
So, (y - 12)/(x - 9) = 1/3
cross-multiplying, we have
3(y - 12) = (x - 9)
expanding the brackets, we have
3y - 36 = x - 9
collecting like terms, we have
3y = x - 9 + 36
3y = x + 27
So, the equation of the radius line is
3y - x = 27.
Now, since the center of the circle lies on the line 2y + x = 8, the radius line and this line intersect at the center of the circle. So, we solve both equations simultaneously to find the center of the circle.
So, 3y - x = 27 (1)
2y + x = 8 (2)
adding (1) and (2), we have
5y = 35
y = 35/5
y = 7
Substituting y = 7 into (2), we have
2y + x = 8
2(7) + x = 8
14 + x = 8
x = 8 - 14
x = -6
So, the center of the circle is at (-6, 7)
b. The radius of the circle
The radius of the circle is the length of the line from the tangent point to the center of the circle. So, r = √[(x₂ - x₁)² + (y₂ - y₁)²] where (x₁, y₁) = (-6, 7) and (x₂, y₂) = (9, 12).
Substituting these into r, we have
r = √[(9 - (-6))² + (12 - 7)²]
= √[(9 + 6)² + (12 - 7)²]
= √[15² + 5²]
= √[225 + 25]
= √250
= √25 × √10
= 5√10 units
c. The equation of the circle
Now, the equation of a circle with center (h,k) and radius, r is
(x - h)² + (y - k)² = r² where (h. k) = (-6,7)
Substituting the variables into the equation, we have
(x - (-6))² + (y - 7)² = (√250)²
(x + 6)² + (y - 7)² = 250
expanding the brackets, we have
x² + 12x + 36 + y² - 14y + 49 = 250
collecting like terms, we have
x² + y² + 12x - 14y + 36 + 49 - 250 = 0
x² + y² + 12x - 14y - 165 = 0
So, the equation of the circle is x² + y² + 12x - 14y - 165 = 0 , its center is (-6, 7) and its radius is 5√10 units.
