Practice Tangent Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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 floorβ€”the floor touches the ball at exactly one point.

Showing a random 20 of 50 problems.

Example 1

medium
Find the value of cc that makes y=x+cy = x + c tangent to x2+y2=8x^2 + y^2 = 8.

Example 2

easy
True or false: a tangent line to a circle intersects the circle at exactly one point.

Example 3

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

Example 4

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).

Example 5

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

Example 6

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.

Example 7

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 8

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

Example 9

medium
Two tangent segments are drawn from an external point PP to a circle, touching at AA and BB. How do lengths PAPA and PBPB compare?

Example 10

medium
Find the slope of the tangent to y=x2y = x^2 at the point (2,4)(2, 4).

Example 11

medium
Determine whether the line y=2x+1y = 2x + 1 is tangent to, secant to, or misses the circle x2+y2=1x^2 + y^2 = 1.

Example 12

hard
How many common tangents do two internally tangent circles share?

Example 13

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

Example 14

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

Example 15

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

Example 16

medium
Find the tangent length from (8,0)(8, 0) to the circle x2+y2=36x^2 + y^2 = 36.

Example 17

medium
Find the equation of the tangent line to the circle x2+y2=169x^2 + y^2 = 169 at the point (5,12)(5, 12).

Example 18

easy
Is the line y=5y = 5 tangent to the circle x2+y2=25x^2 + y^2 = 25?

Example 19

challenge
Prove that the tangent line to the parabola y=x2y = x^2 at the point (a,a2)(a, a^2) has xx-intercept a2\dfrac{a}{2} for any a≠0a \neq 0.

Example 20

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