Tangents to Circles

Tangents to Circles#

Circle Properties Revision#

Theory#

Before studying tangents, let’s review essential circle properties:

  • Circle equation: \((x - h)^2 + (y - k)^2 = r^2\)

  • A tangent touches the circle at exactly one point

  • The tangent is perpendicular to the radius at the point of contact

  • From an external point, two tangents can be drawn to a circle

Application#

For circle \((x - 3)^2 + (y - 4)^2 = 25\):

  • Center: \((3, 4)\)

  • Radius: \(r = 5\)

  • Any tangent at point \(P\) on the circle is perpendicular to line \(CP\)

Tangents to Circles#

Theory#

Tangent at a Given Point on the Circle#

For a circle \((x - h)^2 + (y - k)^2 = r^2\) and point \((x_1, y_1)\) on the circle, the tangent equation is: $\((x_1 - h)(x - h) + (y_1 - k)(y - k) = r^2\)$

For circle \(x^2 + y^2 = r^2\), the tangent at \((x_1, y_1)\) is: $\(x_1x + y_1y = r^2\)$

Tangent with Given Slope#

To find tangents with slope \(m\) to circle \((x - h)^2 + (y - k)^2 = r^2\):

  1. Tangent form: \(y - k = m(x - h) \pm r\sqrt{1 + m^2}\)

  2. Two tangents exist (unless line passes through center)

Tangents from External Point#

From external point \((x_0, y_0)\) to circle \((x - h)^2 + (y - k)^2 = r^2\):

  1. Length of tangent: \(L = \sqrt{(x_0 - h)^2 + (y_0 - k)^2 - r^2}\)

  2. Angle between tangents: \(\theta = 2\sin^{-1}\left(\frac{r}{\sqrt{(x_0 - h)^2 + (y_0 - k)^2}}\right)\)

Interactive Visualization: Tangent Lines Explorer#

Application#

Examples#

Example 1: Tangent at a Point Find the equation of the tangent to circle \(x^2 + y^2 = 25\) at point \((3, 4)\).

Solution: Using the formula \(x_1x + y_1y = r^2\): \(3x + 4y = 25\)

This is the tangent equation.

Example 2: Tangents with Given Slope Find tangent lines to circle \((x - 2)^2 + (y - 1)^2 = 9\) with slope \(m = \frac{4}{3}\).

Solution: Center: \((2, 1)\), radius: \(r = 3\)

Using the formula: \(y - 1 = \frac{4}{3}(x - 2) \pm 3\sqrt{1 + \frac{16}{9}}\) \(y - 1 = \frac{4}{3}(x - 2) \pm 3\sqrt{\frac{25}{9}}\) \(y - 1 = \frac{4}{3}(x - 2) \pm 5\)

Tangent lines:

  • \(y = \frac{4}{3}x - \frac{8}{3} + 1 + 5 = \frac{4}{3}x + \frac{10}{3}\)

  • \(y = \frac{4}{3}x - \frac{8}{3} + 1 - 5 = \frac{4}{3}x - \frac{20}{3}\)

Example 3: Tangents from External Point Find the length of tangents from point \((7, 1)\) to circle \(x^2 + y^2 - 4x - 6y + 9 = 0\).

Solution: First, find center and radius: \((x - 2)^2 + (y - 3)^2 = 4\) Center: \((2, 3)\), radius: \(r = 2\)

Length of tangent: \(L = \sqrt{(7 - 2)^2 + (1 - 3)^2 - 4}\) \(L = \sqrt{25 + 4 - 4}\) \(L = \sqrt{25} = 5\)

Multiple Choice Questions#

Sector Specific Questions: Tangent Applications#

Key Takeaways#

Important

Tangents to Circles - Essential Concepts

  1. Tangent at a Point: For circle \(x^2 + y^2 = r^2\) and point \((x_1, y_1)\):

    • Tangent: \(x_1x + y_1y = r^2\)

  2. Tangent Properties:

    • Perpendicular to radius at contact point

    • From external point: exactly 2 tangents

    • From point on circle: exactly 1 tangent

    • From internal point: no tangents

  3. Length Formula: From external point \((x_0, y_0)\) to circle with center \((h, k)\) and radius \(r\): $\(L = \sqrt{(x_0 - h)^2 + (y_0 - k)^2 - r^2}\)$

  4. Tangents with Given Slope \(m\): $\(y - k = m(x - h) \pm r\sqrt{1 + m^2}\)$

  5. Applications: Belt drives, optical systems, risk analysis, lighting design

Contents