Math Problem Statement
Solution
Let's start by analyzing and solving the problem.
The given circle is: which is a circle centered at the origin with radius 5, since .
Part (a): Find the equation of the tangent line to the circle at the point .
The slope of the radius at the point is the slope from the origin to this point, which is: Since the tangent line is perpendicular to the radius, the slope of the tangent line is the negative reciprocal of , which is: Using the point-slope form of a line equation, , with slope and point , we get: Simplifying: Thus, the equation of the tangent line is:
Part (b): Find the other point on the circle where a tangent line is parallel to the tangent line in part (a).
For the tangent line to be parallel, it must have the same slope, .
We know the equation of the circle is . The slope of the tangent line at any point on the circle is the negative reciprocal of the slope of the radius at that point. The slope of the radius at any point is , so the slope of the tangent line is .
To find the point where the slope of the tangent line is , we set , which gives: Now, substitute this into the equation of the circle : If , then . So one point is .
Thus, the other point where the tangent line is parallel to the one found in part (a) is .
Would you like any further details or clarifications? Here are some related questions:
- How can we find the equation of a tangent line to any circle?
- Why is the slope of the tangent line the negative reciprocal of the slope of the radius?
- How does the geometry of a circle influence the relationship between tangent lines and radii?
- What are other methods to find the points of tangency on a circle?
- How would the solution change if the center of the circle was not at the origin?
Tip: For any circle, the slope of the tangent line at a point can be derived directly from the negative reciprocal of the radius's slope at that point.
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Math Problem Analysis
Mathematical Concepts
Analytic Geometry
Circles
Tangents
Slopes
Formulas
Equation of a circle: x^2 + y^2 = r^2
Slope of a line: (y2 - y1) / (x2 - x1)
Equation of a line: y - y1 = m(x - x1)
Theorems
The tangent to a circle is perpendicular to the radius at the point of tangency.
Suitable Grade Level
Grades 10-12
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