Tangent Intuition Math Example 3

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

Example 3

medium
Verify that the line y=x+52y = x + 5\sqrt{2} is tangent to the circle x2+y2=25x^2 + y^2 = 25, and find the point of tangency.

Solution

  1. 1
    Step 1: Substitute y=x+52y = x + 5\sqrt{2} into the circle: x2+(x+52)2=25โ‡’2x2+102x+50=25โ‡’2x2+102x+25=0x^2 + (x + 5\sqrt{2})^2 = 25 \Rightarrow 2x^2 + 10\sqrt{2}x + 50 = 25 \Rightarrow 2x^2 + 10\sqrt{2}x + 25 = 0.
  2. 2
    Step 2: Discriminant: ฮ”=(102)2โˆ’4(2)(25)=200โˆ’200=0\Delta = (10\sqrt{2})^2 - 4(2)(25) = 200 - 200 = 0. One repeated root โ€” tangent confirmed.
  3. 3
    Step 3: x=โˆ’1024=โˆ’522x = \dfrac{-10\sqrt{2}}{4} = \dfrac{-5\sqrt{2}}{2}, y=x+52=522y = x + 5\sqrt{2} = \dfrac{5\sqrt{2}}{2}.

Answer

Tangent confirmed; point of tangency: (โˆ’522,โ€‰522)\left(-\dfrac{5\sqrt{2}}{2},\, \dfrac{5\sqrt{2}}{2}\right).
A line is tangent to a circle when the system of equations has exactly one solution, i.e. the discriminant equals zero. The repeated root gives the point of tangency.

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 1 medium

Find the equation of the tangent line to the circle [formula] at point [formula].

Example 2 hard

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

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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