# The shortest distance and the y-intercept

In this investigation you will explore how the distance from a point to a line changes as the y-intercept of the line changes.

**Note:**the shortest distance from a point to a line is the perpendicular distance. AD in the diagram below. Also, it will be easier for you to see patterns if**at least**the decimal results for the change in distance you find are written as fractions where possible. ## TASK

The diagram below shows a line with equation, . Sliders have been set up so that the values of the parameters can be changed.
Also, a point A has been placed on the diagram.

## Section A; Level 1 - 4

**1.**For the line shown above,

**DESCRIBE**how the value of the y-intercept is related to the values of b and c?

**2.**To check your understanding of the mathematics involved in this investigation, set a = 6, b = 8, c = -32 and point A as (4, 5). Then, using an

**ALGEBRAIC**method verify that the length of AB is 3.2.

**The steps involved are**;

- find the equation of the line AB
- find where AB intersects the line in order to find the coordinates of point B
- use the distance formula to find the length of AB

**Note:**the distance formula is:

**From now on you will use the diagram above to find the information you need. 3. Set a = -3 and b = 4.**i. Set the point A to (4, 1) and then find the distance AB when the y-intercept is 1, 2, 3, etc.

**Record the y-intercept,**

**distance and change in distance**

**data in a table.**ii. Describe how the distance AB changes as the y-intercept increases -

**write the change in distance as a fraction in simplest form.**ii. Change the position of point A to (5, 3) and again, find the distance AB when the y-intercept is 1, 2, 3, etc. iv. How does changing the position of point A impact the pattern you described in part ii. above?

**4. Set a = -7 and b = 24**i. Set the point A to (4, 0) and again, find the distance AB when the y-intercept is 1, 2, 3, etc. Record the distance and y-intercept data in another table. ii. Describe any patterns in the way the distance changes -

**write the change in distance as a fraction in simplest form.**

**5. Set a = -5 and b = 12**i. Keep the point A at (4, 0) and again, find the distance AB when the y-intercept is 1, 2, 3, etc. Record the distance and y-intercept data in another table.

**Note:**Calculate the change in distance correct to

**at least 7-decimal places**. And, use the decimal to fraction convertor shown below to change to a fraction (you will have to check the

**approximate**box). ii. Describe any patterns in the way the distance changes this time.

## Decimal to fraction convertor - scroll down and use the approximate option.

**6.**By considering the data you have collected so far, and any patterns you have observed, what conjecture can you make about the relationship between the

**CHANGE in the shortest distance from a point to a line**as the y-intercept changes and the values of the parameters a and b when

**Write a RULE that fits your conjecture.**
**
**

**7.**Use your rule to predict the relationship between the

**CHANGE**in distance to the line and the y-intercept when;

**a = -9 and b = 40**?

**8.**Test your prediction from question 7, using the diagram above.

## Section B; Level 5 - 8

**9.**By selecting other values for a and b, explore the situation further. To help you do this try some of the following suggestions: i. look at the examples you have already used and switch the values for

*a*and

*b*around. And, you could also look at what happens when you change the signs of

*a*and

*b*. ii. try setting a = 8 and b = -15, or a = -15 and b = 8, etc iii. try doubling some of the values you have already used for a and b. So where a = -3 and b = 4, try a = -6 and b = 8, etc iv. try setting a = 48 and b = -55, etc

**In this section you MUST show evidence that you have;**

- Selected and applied problem solving techniques to discover
**PATTERNS**in the new data you have collected, - used the new data you collected to decide whether you need to change your previous
**RULE**from Question 6 to write a more general rule, **VERIFY**your general rule