Tangent on a Circle

Tangent of a Circle

A circle with centre (0,0) and radius r has the equation x² + y² = r². Find the tangent to the circle x² + y² = 25 at the point (-3,4):

Step 1

First calculate the gradient of the line from the origin to (-3,4):
- Gradient of purple line is m = ⁴⁄_-3

Step 2

The tangent will be perpendicular to the purple line so the gradient of the tangent is ^-1⁄_m:
- gradient of tangent = ^-1⁄_m = ³⁄₄

Step 3

Substitute in (-3,4) to y = ³⁄₄x + c:
- 4 = -⁹⁄₄ + c
- c = ²⁵⁄₄
- So the equation of tangent is y = ³⁄₄x + ²⁵⁄₄.

Translations & Reflections

Translations are when we move a graph without changing its shape. A reflection is when you reflect the shape in a given line.

Vertical translation

Moves a function up or down.
You can add and subtract from f(x) in the form y = f(x) to produce a vertical translation:
- y = f(x) + a moves the graph up by a
- y = f(x) - a moves the graph down by a

Horizontal translation

Moves a function left or right.
You can add and subtract from x in the form y = f(x) to produce a horizontal translation:
- y = f(x + a) moves the graph left by a
- y = f(x - a) moves the graph right by a

Reflection in the x axis

For a function y = f(x):
- y = -f(x) gives a reflection in the x axis

Reflection in the y axis

For a function y = f(x):
- y = f(-x) gives a reflection in the y axis

Invariant points

Invariant points are points that don’t change in a transformation (e.g reflection).
(0, 1.5) is the invariant point shown in the above reflection.

1Number

1.1Using Numbers

1.2Fractions, Decimals & Percentages

1.2.1Adding Fractions

1.2.2Subtracting Fractions

1.2.3Multiplying Fractions

1.2.4Dividing Fractions

1.2.5Mixed Fractions

1.2.6Calculating Percentages of Numbers

1.2.7Calculating Fractions of Numbers

1.2.8Multiply Decimals

1.2.9Place Value

1.2.10Recurring Decimals

1.2.11Exam-Style Questions - Recurring Decimals

1.2.12Convert Fractions & Terminating Decimals

1.2.13Convert Recurring Decimals

1.2.14Convert Decimals & Percentages

1.2.15Compare Decimals

1.2.16Compare Fractions

1.2.17Exam-Style Questions - Percentages

1.2.18End of Topic Test - Fractions & Decimals

1.3Powers & Roots

1.3.1Powers

1.3.2Roots

1.3.3Indices Advanced

1.3.4Estimating Powers

1.3.5Estimating Roots

1.3.6Multiplying Powers

1.3.7Dividing Powers

1.3.8Power Raised to Power

1.3.9Fractions Raised to a Power

1.3.10Numbers Raised to Negative Powers

1.3.11Roots of Numbers

1.3.12Numbers to the Power of 1 & 0

1.3.13Surds

1.3.14Simplify Surds

1.3.15Rationalise Denominator

1.3.16Exam-Style Questions - Indices

1.3.17End of Topic Test - Powers & Roots

1.4Accuracy

1.4.1Decimal Rounding

1.4.2Significant Figure Rounding

1.4.3Estimation

1.4.4Truncation

1.4.5Limits of Accuracy

1.4.6Bounds Addition

1.4.7Bounds Subtraction

1.4.8Bounds Multiplication

1.4.9Bounds Division

1.4.10End of Topic Test - Rounding & Estimation

2Algebra

2.1Introduction to Algebra

2.1.1Algebra Basics

2.1.2Substituting Values

2.1.3Algebraic Vocabulary

2.1.4Collecting Terms

2.1.5End of Topic Test - Algebra Basics & Power Rules

2.1.6Expanding Brackets I

2.1.7Common Factors

2.1.8Expanding Brackets II

2.1.9Rearrange For x

2.1.10Rearrange with Powers

2.1.11Exam-Style Questions - Basics of Algebra

2.1.12End of Topic Test - Expanding & Rearranging

2.2Manipulating Algebra

2.2.1How to Factorise

2.2.2Factorising Quadratics

2.2.3Algebraic Fractions - Addition & Subtraction

2.2.4Algebraic Fractions - Multiplication & Division

2.3Proofs & Functions

2.3.1Identities & Proofs

2.3.2Exam-Style Questions - Algebraic Proof

2.3.3End of Topic Test - Algebra & Proofs

2.3.4Functions

2.3.5Combine Functions

2.3.6Composite Functions

2.3.7Inverse Functions

2.4Straight Line Graphs

2.4.1Straight Line Graphs

2.4.2y = mx + c

2.4.3Grad + YIntercept

2.4.4Exam-Style Questions - Straight Line Graphs

2.4.5End of Topic Test - Functions & Straight Lines

2.5Common Graphs

2.5.1Shape of Quadratic Graphs

2.5.2Roots of a Quadratic Graph

2.5.3Completing Squares

2.5.4Turning Points

2.5.5Common Functions 1

2.5.6Common Functions 2

2.5.7Trigonometric functions

2.5.8End of Topic Test - Common Graphs

2.6Transformations & Tangents

2.6.1Transformations

2.6.2Tangent on a Circle

2.7Properties of Graphs

2.7.1Estimate Gradient

2.7.2Estimate Area

2.7.3Represent Gradient & Area

2.7.4Distance-Time Graph

2.7.5Velocity-Time Graph

2.7.6End of Topic Test - Transformations & Graphs

2.8Solving Equations

2.8.1Solve Linear Equations

2.8.2Linear Equations with Fractions

2.8.3Solve Linear Equations Graphically

2.8.4Quadratic ±

2.8.5Factorise Quadratics

2.8.6Quadratic Formula

2.8.7Complete the Square

2.8.8Solve Quadratics Graphically

2.8.9End of Topic Test - Solving Equations

2.8.10Simultaneous Equations

2.8.11Quadratic Simultaneous Equations

2.8.12Simultaneous Equations - Graphs

2.8.13Simultaneous Equations - Word Problems

2.8.14Approximate Solutions - Number of Iterations

2.8.15Approximate Solutions - Iterations to 1 DP

2.8.16Approximate Solutions - Iteration Formula

2.8.17Exam-Style Questions - Simultaneous Equations

2.8.18End of Topic Test - Simultaneous Equations

2.9Inequalities

2.9.1Solving Inequalities

2.9.2Solving Double Inequalities

2.9.3Solving Quadratic Inequalities

2.9.4Representing Inequalities

2.9.5Exam-Style Questions - Inequalities

2.9.6End of Topic Test - Inequalities

2.10Sequences

2.10.1Nth Term Rules

2.10.2Recurrence Relationships

2.10.3Find Nth Term in Linear Sequence

2.10.4Check If a Number is In a Sequence

2.10.5Exam-Style Questions - Sequences

2.10.6Arithmetic & Geometric Progressions

2.10.7Key Sequences

2.10.8Quadratic Sequences

2.10.9Find Nth Term in Quadratic Sequence

2.10.10Exam-Style Questions - Geometric Series

2.10.11End of Topic Test - Sequences

3Ratio

3.1Ratios in Practice

3.1.1Converting Between Units

3.1.2Formula Triangles

3.1.3Scale Factors

3.1.4Ratios in Shapes

3.2Manipulating Ratios

3.2.1Simplifying Ratios

3.2.2Turn Ratios into Equations

3.2.3Ratio Notation & Reduction

3.2.4End of Topic Test - Units & Ratios

3.2.5Part : Part Ratio

3.2.6Part : Whole Ratio

3.2.7Two Ratios

3.2.8Differences in Ratios

3.2.9Fractions in Ratios

3.2.10Numbers as Fractions

3.2.11Ratios to Fractions

3.2.12Exam-Style Questions - Ratios

3.2.13End of Topic Test - Fractions & Ratios

3.3Percentage & Interest

3.3.1Convert Percentages to Decimals

3.3.2Convert Percentages to Fractions

3.3.3Find the Percentage

3.3.4More/Less Percentage

3.3.5Price from Percentage Change

3.3.6Calculate Percentage Change from Price

3.3.7End of Topic Test - Percentages

3.3.8Profit & Interest & Discount

3.3.9Original Price after Discounts

3.3.10Simple Interest

3.3.11Compound Growth & Decay

3.3.12Compound Interest

3.3.13Compound Interest - After How Many Days?

3.3.14End of Topic Test - Percentages in Finance

3.4Proportion

3.4.1Proportion

3.4.2Direct Proportion

3.4.3Inverse Proportion

3.4.4Exam-Style Questions - Proportionality

3.5Gradient

3.5.1Gradient on a Straight Line

3.5.2Direct & Inverse Proportions as Graphs

3.5.3Gradient of a Curve

3.5.4Positive & Negative Gradients

3.5.5End of Topic Test - Proportion & Gradient

4Geometry

4.1Introduction to Geometry

4.1.1Geometry Basics

4.1.2Geometry Basics 2

4.1.3Regular Polygons

4.1.4Triangles & Angles

4.1.5Midpoint

4.1.6Bisectors & Perpendiculars

4.1.7Locus

4.1.8Angle rules

4.1.9Parallel Angle Rules

4.1.10Angles in a Polygon

4.1.11Angles in a Triangle

4.1.12Exam-Style Questions - Angles in Shapes

4.1.13End of Topic Test - Geometry

4.2Triangles & Quadrilaterals

4.2.1Types of Triangle

4.2.2Angles & Triangles

4.2.3Congruence

4.2.4Similarity

4.2.5Quadrilaterals

4.3Transformations

4.3.1Types of Transformations

4.3.2Invariant Points

4.3.3Effects of Transformations

4.3.4Exam-Style Questions - Reflection

4.4Circle Basics

4.4.1Properties of a Circle 1

4.4.2Chords

4.4.3Properties of a Circle 2

4.4.4Area of a Circle

4.4.5Circumference of a Circle

4.4.6Fractions of Circles

4.4.7Exam-Style Questions - Circle Formulae

4.4.8End of Topic Test - Triangles & Circles

4.5Circle Theorems

4.5.1Perpendicular Bisector of a Chord

4.5.2Angle at the Centre

4.5.3Angle of a Semicircle

4.5.4Opposite Angles in Cyclic Quadrilaterals

4.5.5Tangent Lengths

4.5.6Angle in the Same Segment

4.5.7Alternate Segment Theorem

4.5.8Exam-Style Questions - Circle Theorems

4.6Measurements & Units

4.6.1Length

4.6.2Area

4.6.3Volume

4.6.4Capacity

4.6.5Time

4.6.6Money

4.6.7Mass

4.6.8Bearings

4.6.9Scale Factor & Enlargement

4.6.10Exam-Style Questions - Enlargements

4.7Calculating Area

4.7.1Area of 2D Shapes

4.7.2Exam-Style Questions - Area

4.7.3End of Topic Test - Circle Theorems, Units & Area

4.8Triangle Formulae

4.8.1Pythagoras' Theorem

4.8.2Pythagoras in 3D

4.8.3Trigonometry

4.8.4Exact Values

4.8.5Sine Rule

4.8.6Cosine Rule

4.8.7Area of a Triangle

4.8.8Exam-Style Questions - Trigonometry & Area

4.93D Shapes

4.9.1Types of 3D Shape

4.9.2Plans & Elevations

4.9.3Nets

4.9.4Volume of 3D Shapes

4.9.5Volume of a Cuboid

4.9.6Volume of a Cylinder

4.9.7Volume of a Sphere

4.9.8Volume of a Hemisphere

4.9.9Volume of a Cone

4.9.10Volume of a Pyramid

4.9.11Surface Area & Composite Shapes

4.9.12Surface Area of a Cuboid

4.9.13Surface Area of a Cylinder

4.9.14Surface Area of a Sphere/Hemisphere

4.9.15Surface Area of a Cone

4.9.16Exam-Style Questions - Volume

4.10Vectors

4.10.1Vectors Introduction

4.10.2Vectors

4.10.3Translation

4.10.4Scalar

4.10.5Exam-Style Questions - Vectors

4.10.6End of Topic Test - Trigonometry, Shapes & Vectors

5Probability

6Statistics

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