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Shortest Distance between a Point and a Circle

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Shortest Distance between a Point and a Circle

Study Guide

Key Definition

The shortest distance between a point $P(x_1, y_1)$ and a circle with center $(h, k)$ and radius $r$ is $\left| \sqrt{(x_1 - h)^2 + (y_1 - k)^2} - r \right|$.

Important Notes

The distance formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ is essential.
The shortest distance can be zero if the point lies on the circle.
For circles centered at the origin, the formula simplifies.
If the point is inside the circle, the distance is still valid.
Use absolute value to ensure non-negative distance.

Mathematical Notation

$\sqrt{x}$ represents the square root of $x$

$+$ represents addition

$-$ represents subtraction

Remember to use proper notation when solving problems

Why It Works

The formula calculates the straight-line distance from the point to the circle, considering the circle's curvature and the point's position.

Remember

Always use the formula $\left| \sqrt{(x_1 - h)^2 + (y_1 - k)^2} - r \right|$ for finding the shortest distance.

Quick Reference

Distance Formula:$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Understanding Shortest Distance between a Point and a Circle

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The shortest distance from a point $(x_1,y_1)$ to a circle with center $(h,k)$ and radius $r$ is found by first measuring the straight-line distance to the center using $\sqrt{(x_1-h)^2+(y_1-k)^2}$, then subtracting $r$, and finally taking the absolute value: $|\sqrt{(x_1-h)^2+(y_1-k)^2}-r|$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the shortest distance from the point $(3, 4)$ to the circle $x^2 + y^2 = 9$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are plotting a map where a circle represents a park with center $(1, 2)$ and radius $4$, and a point represents your home at $(5, 6)$. Calculate the shortest path from your home to the park's boundary using the distance formula.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a circle with center $(2, -1)$ and radius $5$, find the shortest distance to the point $(6, 3)$.

Challenge Quiz

Single Choice Quiz

Advanced

Find the shortest distance between the circle $(x - 1)^2 + (y + 2)^2 = 16$ and the point $(5, -6)$.

Recap

Watch & Learn

Review key concepts and takeaways