Math Problem Statement

find the equation tangent to the circle with centre (1, 2) and radius V5 at the point (3, 3)

Solution

To find the equation of the tangent to the circle at the given point, we can proceed as follows:

  1. Identify the Circle's Equation: The general equation of a circle with center (h,k)(h, k) and radius rr is: (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2 Given that the center is (1,2)(1, 2) and radius 5\sqrt{5}, the equation of the circle is: (x1)2+(y2)2=5(x - 1)^2 + (y - 2)^2 = 5

  2. Find the Slope of the Radius at Point (3,3)(3, 3): The radius to the point (3,3)(3, 3) has a slope given by: mradius=y2y1x2x1=3231=12m_{\text{radius}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{3 - 1} = \frac{1}{2}

  3. Determine the Slope of the Tangent Line: The tangent line at a point on a circle is perpendicular to the radius at that point. Therefore, the slope of the tangent mtangentm_{\text{tangent}} is the negative reciprocal of the radius’s slope: mtangent=112=2m_{\text{tangent}} = -\frac{1}{\frac{1}{2}} = -2

  4. Write the Equation of the Tangent Line: Using the point-slope form of a line, yy1=m(xx1)y - y_1 = m(x - x_1), with m=2m = -2 and passing through (3,3)(3, 3), we get: y3=2(x3)y - 3 = -2(x - 3) Simplifying this: y3=2x+6y - 3 = -2x + 6 y=2x+9y = -2x + 9

Therefore, the equation of the tangent line to the circle at the point (3,3)(3, 3) is: y=2x+9y = -2x + 9

Would you like more details on any step, or do you have questions about this process?

Related Questions

  1. How do you find the equation of a tangent to a circle at any given point?
  2. What are the conditions for two lines to be perpendicular?
  3. How can the distance from a point to a line be calculated?
  4. What is the geometric significance of a tangent line to a circle?
  5. How would this change if the radius or center of the circle were different?

Tip

Remember, for a tangent line to a circle, always find the slope of the radius at the tangent point and then use the negative reciprocal for the tangent's slope.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Circle Equation
Slopes of Lines
Tangent to a Circle

Formulas

Equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Slope formula: (y2 - y1) / (x2 - x1)
Point-slope form of a line: y - y1 = m(x - x1)

Theorems

The perpendicularity of tangent and radius

Suitable Grade Level

Grades 10-12