Tangent Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Tangent Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

A basketball resting on a flat floor—the floor touches the ball at exactly one point.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A tangent line grazes a curve at exactly one point and points the same way the curve does there.

Common stuck point: The procedure for tangent intuition is the easy part; the trap is calling a two-point crossing line a tangent. Asking "Does this line touch the curve at exactly one point and share the curve's direction there?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this line touch the curve at exactly one point and share the curve's direction there?

Worked Examples

Example 1

medium

Find the equation of the tangent line to the circle

x^2 + y^2 = 25

at point

P(3, 4)

Answer

3x + 4y = 25

First step

Step 1: The radius to

P(3,4)

has slope

m_r = \dfrac{4-0}{3-0} = \dfrac{4}{3}

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Example 2

hard

From external point

Q(7, 0)

, find the length of the tangent to the circle

x^2 + y^2 = 25

Tangent from external point Q(7, 0) to the circle; tangent length = 2√6, radius = 5.

Example 3

easy

A tangent from external point

P

touches a circle at

T

. If

|OT| = 5

(radius) and

|OP| = 13

, find

|PT|

Example 4

medium

Find the equation of the tangent line to the circle

x^2 + y^2 = 169

at the point

(5, 12)

Example 5

medium

Find the slope of the tangent to

y = x^2

at the point

(2, 4)

Example 6

medium

Find the value of

c

that makes

y = x + c

tangent to

x^2 + y^2 = 8

Example 7

medium

Find the equation of the tangent to the parabola

y = x^2

at the point

(-1, 1)

Example 8

hard

Find the equations of the two tangent lines from

(0, 6)

to the circle

x^2 + y^2 = 9

Example 9

hard

Find the slope of the tangent to the ellipse

\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1

at the point

(2, \tfrac{3\sqrt{3}}{2})

Example 10

hard

Find the slope of the tangent line to

y = \sin x

x = \dfrac{\pi}{3}

Example 11

challenge

Prove that the tangent line to the parabola

y = x^2

at the point

(a, a^2)

has

x

-intercept

\dfrac{a}{2}

for any

a \neq 0

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Verify that the line

y = x + 5\sqrt{2}

is tangent to the circle

x^2 + y^2 = 25

, and find the point of tangency.

Example 2

easy

At what angle does the tangent to a circle meet the radius drawn to the point of tangency? Use this to explain why a tangent touches the circle at only one point.

The tangent meets the radius at exactly 90° — it touches the circle at only one point.

Example 3

easy

How many points does a tangent line share with a curve at the point of tangency?

Example 4

easy

y = 1

tangent to the circle

x^2 + y^2 = 1

? If so, where?

Example 5

easy

A tangent line to a circle is perpendicular to what at the point of tangency?

A tangent is always perpendicular to the radius at the point of tangency.

Example 6

easy

Does a tangent line match the curve's direction at the touch point?

Example 7

easy

A basketball rests on a flat floor. The floor is tangent to the ball at how many points?

Example 8

easy

A line crosses a curve passing through two points. Is it a tangent or a secant?

Example 9

easy

What is the slope of the tangent to a horizontal curve at its lowest point (a valley bottom)?

Example 10

easy

True or false: a tangent line can never touch the curve again elsewhere.

Example 11

medium

A tangent touches the circle

x^2 + y^2 = 25

(3, 4)

. Find the slope of the tangent.

Example 12

medium

Why is a tangent line the 'limit' of secant lines?

Example 13

medium

A circle has center

(0,0)

and radius 5. Write the equation of the tangent line at

(0, 5)

Example 14

medium

From an external point, how many tangent lines can be drawn to a circle?

Example 15

medium

Two tangent segments are drawn from an external point

P

to a circle, touching at

A

and

B

. How do lengths

PA

and

PB

compare?

Example 16

medium

Why does a larger circle look 'flatter' near any point, in terms of how its tangent approximates it?

Example 17

medium

At a peak (local maximum) of a curve, what is the tangent line's slope?

Example 18

medium

A tangent line to a curve at a point is the best straight-line approximation there. What does this let you do for small steps?

Example 19

challenge

Find the equation of the tangent to the circle

x^2 + y^2 = 25

at the point

(3, 4)

Example 20

challenge

A tangent segment from external point

P

has length 12, and the circle's radius is 5. Find the distance from

P

to the center.

Right triangle: legs 5 and 12, hypotenuse = center-to-P distance = 13.

Example 21

challenge

Explain why the discriminant of the substituted equation tells you if a line is tangent to a circle.

Example 22

challenge

Why does the tangent line's slope at a point on a curve give the curve's instantaneous rate of change there?

Example 23

easy

What angle does a tangent line make with the radius drawn to the point of tangency?

Example 24

easy

Is the line

x = 3

tangent to the circle

x^2 + y^2 = 9

Example 25

easy

The slope of the tangent to a smooth curve at a local maximum is ___.

Example 26

easy

How many common tangent lines do two non-overlapping circles (externally disjoint) have?

Example 27

medium

Find the tangent length from

(8, 0)

to the circle

x^2 + y^2 = 36

Example 28

medium

Determine whether the line

y = 2x + 1

is tangent to, secant to, or misses the circle

x^2 + y^2 = 1

Example 29

medium

From external point

P

, two tangents to a circle have lengths

7

. The angle between them is

60°

. Find the distance from

P

to the center.

Example 30

medium

Two circles, radii

3

and

5

, have centers

10

apart. How many common tangents exist?

Example 31

medium

A line is tangent to

x^2 + y^2 = 4

at the point

(\sqrt{2}, \sqrt{2})

. Find its equation.

Example 32

hard

The line

y = mx - 5

is tangent to the circle

x^2 + y^2 = 5

. Find all values of

m

Example 33

hard

Two circles of radius

4

are externally tangent. How long is the common external tangent segment between their points of contact?

Example 34

hard

The tangent at

(3, 4)

x^2 + y^2 = 25

meets the

x

-axis at what point?

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Related Concepts

Line Circles Derivative Tangent Line

Background Knowledge

These ideas may be useful before you work through the harder examples.

linecircles