Theory and Examples

Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

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Identify the points where the circle crosses the coordinate axes by setting \(y=0\) to find the x-intercepts and \(x=0\) to find the y-intercepts in the circle equation \(\left(x - 2\right)^2 + \left(y - 1\right)^2 = 5\).

Solve for the x-intercepts by substituting \(y=0\) into the equation and solving for \(x\), and solve for the y-intercepts by substituting \(x=0\) and solving for \(y\).

For each intercept point found, calculate the derivative \(\frac{dy}{dx}\) implicitly from the circle equation to find the slope of the tangent line at that point. Use implicit differentiation on \(\left(x - 2\right)^2 + \left(y - 1\right)^2 = 5\).

Use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point of tangency and \(m\) is the slope found from the derivative, to write the equation of the tangent line at each intercept.

Simplify the tangent line equations to standard or slope-intercept form as needed to express the final equations of the tangents at the points where the circle crosses the coordinate axes.

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주요 개념

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Equation of a Circle

The equation (x − h)² + (y − k)² = r² represents a circle with center (h, k) and radius r. Understanding this form helps identify the circle's position and size, which is essential for finding points on the circle, such as where it intersects the coordinate axes.

Finding Points of Intersection with Coordinate Axes

To find where a circle crosses the coordinate axes, set x = 0 to find y-intercepts and y = 0 to find x-intercepts. Solving these equations gives the exact points on the circle that lie on the axes, which are needed to determine tangent lines at those points.

Equation of the Tangent Line to a Circle

The tangent line to a circle at a point on its circumference is perpendicular to the radius drawn to that point. Using the slope of the radius, the tangent's slope is the negative reciprocal. The tangent line equation can then be found using point-slope form with the tangent point.

Theory and ExamplesTangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.