Write The Standard Equation Or The Circle: Center At C(3,4), Tangent To The Line Y=1/3x - 1/3 And The Solution Please :(

Write the standard equation or the circle: Center at C(3,4), tangent to the line y=1/3x - 1/3 and the solution please :(

 

Answer:

(x - 3)² + (y - 4)² = 10

Step-by-step explanation:

Center-Radius Standard Equation:

(x - h)² + (y - k)² = r²

Center (h,k): (3,4)

Radius (r): Unknown

Equation of line tangent to the circle: y=(1/3)x - 1/3

Point of Tangency: Unknown

The radius of the circle is perpendicular to the line tangent to the same circle.  The product of theirs slopes is -1.

Step 1: Find the equation passing through the center (3, 4) perpendicular to tangent line:

Given tangent line:

y = (1/3)x - 1/3

Slope (m₁) of the tangent line = 1/3

Slope (m₂) of the equation of line passing through the center:

(m₂)  (1/3) = -1

m₂ = (3/1)(-1)

m₂ = -3  ⇒  slope of the other line

The equation of the other line with slope = -3 passing through center (3,4):

y-y₁ = m(x-1)

y - 4 = -3(x - 3)

y = -3x + 9 + 4

y = -3x + 13

2) Find the point of tangency of the circle and the tangent line by solving the linear system:

Equation (a): y = (1/3)x - 1/3

Equation (b): y = -3x + 13

Equate in terms of x:

(1/3)x - 1/3 = -3x + 13

(3) (1/3)x - 1/3 = -3x + 13(3) ⇒  multiply each term by their LCD 3

x - 1 = -9x + 39

x + 9x = 39 + 1

10x/10 = 40/10

x = 4

Substitute 4 to x:

y = -3x + 13

y = -3(4) + 13

y = -12 + 13

y = 1

The point of tangency or intersection of the the circle and tangent line is (4, 1).

Step 3: Find the radius using the distance formula:

Distance = Radius = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }

Point (x₁, y₁): (3,4)

Point (x₂, y₂): (4,1)

Radius = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }

Radius = \sqrt{(4-3)^{2}+(1-4)^{2} }

Radius = \sqrt{(1)^{2}+(-3)^{2} }

Radius = √(1+9)

Radius = √10

Step 4:  Write the standard form:

Center (h,k): (3,4)

Radius: √10

(x - 3)² + (y - 4)² = (√10)²

(x - 3)² + (y - 4)² = 10

To visualize the circle (x - 3)² + (y - 4)² = 10, the tangent line y = (1/3)x - 1/3, the equation of the line passing through the center and radius y = -3x + 13, the center (3, 4) and the point of tangency (4,1), please click the image below.


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