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Determine the coordinates of the point $P$ which divides the line segment $A B$ where $A$ is the point $(2,4)$ and $B$ is the point $(5,12)$ so that $A P / P B=2 / 3$.

$$(16 / 5,36 / 5)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:46

Find the coordinates of a …

05:08

Find the coordinates of th…

01:14

06:42

Find the point on the line…

04:43

Let $P$ be the point (2,3,…

05:47

If $A$ and $B$ are $(-2,-2…

03:49

Find the coordinates of po…

So in the given question, we are told to find, we are told that there is a line that is joining the points A -2 to and B 28 And now we are told to find the points that divide the coordinates of the point. That divide this line segment into four equal parts Into four equal parts. Right? So in order to divide this line segment into four equal parts, we need three points on this line segment, right? So we need to find each of the coordinates of age of this point. So let's start with the point that is in the middle of this line. Right? So this is the second point at the midpoint of the spine, A. B. As well as those points. So what we can do is we can use the midpoint formula in order to find the coordinates of this point midpoint formula according to the midpoint formula. If we have a line joining the points X one by one and extra way to the mid point of this line, the mid point of this line, we'll have the coordinates X one plus X two divided by two and Y one plus by two divided by two. So over here what we can take as we can take the point, let's say this point is C. D. Right? So they can take C. D. Yes, equal to minus two Plus two, divided by two And 2-plus 8 divided by two. So they might find off this line segment would be it zero two plus eight is 10, 10. They were by two was five. So 05 is the coordinator of the Midpoint of this line segment. And we have this uh this as 05. Right? So now we need to find two points. Right? So when we look at this nine segment, we can see that This point over here is actually the midpoint of the line segment joining c. d. and the .8. Right? So again we can use them in point from lower here, assuming that this point is this point can be taken as E F. Right? So EFS then the midpoint between the .80 and the point coordinates city. Right? So then what we would have is minus two plus zero by two, coma two Plus 5 x two. And this would be equal to minus two by two, which is minus one coma. We have two plus five divided by two, which is seven by two. So E f s seven by two. Right? Even EF -1, 7 x two. And lastly, we have this point over here which which is actually the midpoint of the line segment joining the point C D and B. Right? So in the same way we can take this point as a B C D E. F. Yeah, we can take this point as G. H. Right? So G H. S. Then estimate point off the point B and C. D. So it can be found us zero plus two divided by two and five plus eight divided by two, which would be equal to zero plus two is equal to two. So two rewarded by two is one and five plus eight is 13. So the y coordinate is 13 divided by two. So now we can list all the required all the points that divides this nine segment into four equal parts. Right? So it is a year CD. And G H. The H is the .1, 13 x two. CD is the point zero comma five and E. F. Is the point minus one comma. We have seven by two -1, Coma seven x 2. So these are the required points. I hope you understood the method. Thank you.

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