Tangent Intuition Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
Find the equation of the tangent line to the circle x2+y2=25x^2 + y^2 = 25 at point P(3,4)P(3, 4).

Solution

  1. 1
    Step 1: The radius to P(3,4)P(3,4) has slope mr=4−03−0=43m_r = \dfrac{4-0}{3-0} = \dfrac{4}{3}.
  2. 2
    Step 2: A tangent is perpendicular to the radius at the point of tangency, so tangent slope mt=−34m_t = -\dfrac{3}{4}.
  3. 3
    Step 3: Tangent through P(3,4)P(3,4): y−4=−34(x−3)⇒y=−34x+94+4=−34x+254y - 4 = -\dfrac{3}{4}(x - 3) \Rightarrow y = -\dfrac{3}{4}x + \dfrac{9}{4} + 4 = -\dfrac{3}{4}x + \dfrac{25}{4}.
  4. 4
    Step 4: Equivalently: 3x+4y=253x + 4y = 25.

Answer

3x+4y=253x + 4y = 25
A tangent to a circle at any point is perpendicular to the radius at that point. This is the key intuition: the tangent line 'just touches' the curve, and touching means the radius and tangent are at right angles.

About Tangent Intuition

A line that just barely touches a curve at exactly one point without crossing it, matching the curve's direction at that point.

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More Tangent Intuition Examples

Example 2 hard

From external point [formula], find the length of the tangent to the circle [formula].

Example 3 medium

Verify that the line [formula] is tangent to the circle [formula], and find the point of tangency.

Example 4 easy

At what angle does the tangent to a circle meet the radius drawn to the point of tangency? Use this

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