Friday, May 20, 2022

# Lines ( Introduction to Analytic Geometry)

Fourth late and fourth one now let's talk about the basic formulas we need to know hi guys it's me against our girls welcome back to our growing youtube channel that's my bhagavad-gita think deeper not subscribe welcome .

now for today's discussion let's talk about the first topic in analytic geometry which is lines okay let's talk about this oh this is how it looks like first let's talk about the very basic definition of a line second one let's talk about the general .

Equation of a line third let's talk about the difference between lines rays line segment okay fourth one let's talk about the very basic formulas we need to know in studying lines like the distance formula midpoint of a line segment and lastly the slope of course .

There are lots of um formulas we need to know about lines but let us just focus on that three in this topic okay and lastly let's talk about the four basic equations of a line first one the slope intercept form two point form the point slope form and lastly the intercepts form okay at the end of this discussion let's try to solve some few .

Exercises i will be giving you an assignment where in fact i'll be giving a cash price or or a load and lastly a summary of this discussion so without further ado let's have this and wait wait wait wait if mention a haha anchors a 30-minute video then maybe you can check the description below so that you will know where to .

Skip if gusto that is fine with me you can answer it right away and claim your face thank you now let's talk about the very basic definition of a line so by definition at least two points will determine a line so you have two points you will connect them then you will have a line simple as that and also according to its .

General equation the general equation of a line is a x plus b y plus c is zero where a is the coefficient of x b is the coefficient of y and c is any constant any real number okay now let's talk about lines race and line segment without further ado let's have this okay let's have this okay by definition .

Of a line two points will determine a line so you have two points you connect them then you already have a line okay by the way in a line there are two arrows okay two arrows will signify that it will extend up to infinity okay it will continue to this it will continue to this it will not stop okay tata gaussian whiteboard .

Nothing on the other hand we have this ray okay when you say re it is a semi infinite line because we know where it's we know where it starts but it will continue up to infinity okay and lastly we have this line segment okay meaning it is just a segment of a .

Line it is just a part of a line so it could be this part could be this part it could be this one or it could be this one again it is just a segment of a line okay and now let's talk about the distance formula so distance formula is square root of x2 minus x1 squared plus y2 minus y1 .

Squared okay in the distance formula it is derived from the pythagorean theorem okay the right triangle so let's assume this is the line segment and we wish to find d which is the distance or the length of this line segment .

So let's assume this is point x one and this is point x2 and let's assume this is also point y1 and this is point y2 now this length x okay .

Length x is just equivalent to x2 minus x1 okay and on the other hand y this length y okay this one is just equivalent to y2 minus y1 so assuming d is the hypotenuse so d is just equivalent to x squared plus .

Y squared where x x and y is x sub 2 minus x sub 1 squared plus y sub 2 minus y sub 1 quantity squared so it will just go back to here so pythagorean theorem .

Proof of this formula now let's have an example so say for example we have point b which is three and five and point q which is let's say 10 and seven so we wish to find point .

Or line segment the distance of line segment pq so again by definition d is just equivalent to x sub 2 minus x sub 1 this is x sub 2 x sub 1 10 minus 3 squared plus y sub 2 minus y sub 1 .

7 minus 5 quantity squared so 10 minus 3 is 7 7 squared plus 2 squared so 7 squared is 49 plus 4 which is equivalent to square root of .

53 units okay don't forget to put units because it is a distance it is a length so final answer okay now let's have this midpoint of a line segment okay when you say midpoint .

It is just the middle point of this line segment which is denoted as capital m x bar and y bar now say for example we have this line segment pq with the points x1 and x1 y1 and x2 y2 and the midpoint which is something in between of this p q which is say for .

Example this point we denote that as x bar and y bar now to solve the x bar and y bar all we have to do is just get the average of the two x two plus x one over two and lastly y bar is just y 2 plus y 1 over 2. okay now let's have an example say for example .

P is 7 and 10 and q is four and five four now to get the x bar is x one plus x two which is x seven plus four over two .

Which is seven plus four is eleven so 11 over 2. now how about y bar y bar is just y 2 plus y 1 10 plus 5 or 5 plus 10 over 2 which is 15 over two so therefore the middle point .

Is 11 to 15 over two okay so madaleela now let's talk about slope of a line okay .

When you say slope of line it measures how steep the line is so when you say slope it is how tilted your line at and it is denoted as small letter m okay by equation m is just y2 minus y1 over x2 minus x1 also known as the rice which is the y run which is the x y x and also we can .

Solve for m using tangent theta where theta there is the angle between the horizontal axis and our line so the angle between them so y tangent theta okay according to trigonometry if we will review it tangent theta is just the opposite over adjacent so what is the opposite of theta which is the y okay which is the .

Rise and the adjacent of theta is the x which is the run still it will follow rise over one okay take note of this two equations okay how about this i have two um now i have few reminders from you okay we have the so-called angle of .

Inclination and angle of depression so when you say angle of inclination it is the angle above the horizontal or the horizon and the angle of depression is the angle below the horizon or the horizontal axis so .

Paramount so when you are feeling depressed you are feeling low so depression low okay okay how about this if we have a horizontal line automatically the slope is zero and if we have a vertical line the slope .

Is undefined automatic yeah take note of that okay okay okay moving on and now let's talk about equation of a line okay in this chapter i will be introducing to you four equations of align first one we'll be talking about slope .

Intercept form we are given a slope and y intercept that's why slope intercept so it is in the form y is equivalent y is equal to m x plus b where m is the slope and b is the y intercept so take note of this next one is we have this point slope form we are given a point and a slope so it is in the form y minus y sub 1 .

Is equivalent to m times x minus x sub 1 where x1 and y1 are the point okay next one we have this two point form we are given two points okay the long points it is in the form y minus y sub one equals y two minus y one over x two minus x one .

Times quantity x minus x sub one take note say point slope form and see two point four maka para hala ito minus y one over x two minus x sub one is just this slope it's just that hindi directly slope and vinegar is attend but rather two points .

Where we can still solve the value of m and lastly we have this intercept form okay when you say intercept form we are given the x-intercept and the y-intercept so it is in a form x over a plus y over b is equal to one where a is the x-intercept and b is the .

Y-intercept under x is the x-intercept under y is the y-intercept equal one so a is the x-intercept and b is the y-intercept now let's have an example one by one okay let's go now let's talk deeper about slope intercept form where it is in the form y is equal to .

Mx plus b where m is the slope b is the y-intercept now given these parameters we will try to make an equation out of this we will make it as a slope intercept form so solution for a since the slope and the .

Y-intercept are already given all we have to do is just plug in the values so y mx plus b is just y 5x plus seven .

Okay easy as that how about this in b we are given the general equation of a line a x plus b y plus c is equal to zero now all we have to do is just manipulate this to form it like this so manipulating so 2y is just equivalent to .

Negative 4x plus six okay now dividing both sides by two i'm gigging y is just equivalent to negative two x plus three where the slope is negative two and the y-intercept is .

Positive three okay lastly okay same as well general equation convert into slope intercept form so by manipulation negative 10 y equals negative seven x .

Minus five now dividing both sides by negative 10 mcgee y is equivalent to 7 over 10 x plus one half where the slope is seven over ten .

And the y intercept is one half okay very easy now let's have this let's talk about two-point form and point-slope form against the two-point form at the point-slope form .

Okay difference it's just that c2 point form is measurement see point slope form is direct given slope or it depends on the given situation now say for example we are given two points p and q p is 7 6 and q is 10 and 5 .

Okay out of this given we will make two equation okay so solution okay y minus y sub 1 is equivalent to y2 minus y1 over x2 minus x1 times x minus x sub 1. okay again before i proceed for my solution in choosing which is x1 x2 y1 .

And y2 in this you can choose any any of these two p of q all you have to do is just be consistent in your solution okay be consistent with regards to your assignment okay so say for example my y one is six so y minus six equivalent to y two .

Five minus six over ten minus seven times x minus seven if you choose q as your x one and y one then there's nothing wrong we will be having same output at the end now in here we already have the .

Two point form okay this is the 2.4 now if we wish to combine this or to simplify this y minus 6 equivalent to negative 1 over 3 times x minus 7 then we will be having the point slope form from two-point form combining this we .

Will be having the point-slope form where the m the slope is negative one-third okay now if we wish to combine this more okay further simplification okay we will be having the general equation so let's have this y minus six is equivalent to negative one third times x minus seven multiply three both sides .

So we will get rid of this three so i'm gigging 3y minus 18 i'm gigging negative x okay be careful plus 7 now x plus 3y .

Negative 18 transform seven to here again minus seven negative eighteen minus seven making negative 25 is equal to zero so this will be the general equation of the line ax plus b y plus c is equal to zero so from two point four my gig in point .

Slope form and further simplification my gig general equation okay okay and lastly let's talk about the intercepts form okay the not so common form okay out of the four .

Where it is in the form x over a plus y over b equals to one where a and b must not be equal to zero and a and b are the x and y intercepts respectively okay let's try this let's try to solve this given this graph let's form an equation out of this .

So of course let's put the equation x over a plus y over b equals one the x x-intercept here is the point seven zero when you say x-intercept the point where it touches the x-intercept x-axis rather and of course the y-intercept the point where it touches the y-axis .

So this is the x-intercept four it touches the x-axis so x over seven plus y over what is the y-intercept five five equals one okay .

Okay and take note when you say x intercept automatically y is zero and when you say y intercept the x is zero now how about number two okay number two this is an answer for number one and in number two we are given a general equation where the x and y-intercepts are not given so what we are going to do is of .

Course let us all force let us solve for the x and y intercept or the a and b so for solving the x intercept all we have to do is equate y into zero why it is because if it touches the x axis the value of y is zero and if it touches the y axis the value .

Of x is automatically zero take note of that so in finding x-intercept we let y is equivalent to zero so this will become zero so three x minus six equals zero three x equals six so x is equivalent to two .

So the x intercept is two zero on the other hand in solving for the y intercept we let x is zero since the value of x in the y axis is zero so three x plus four y minus six is zero .

So this is zero since the value of x is zero so four y minus six is zero four y equals six y is just six over four which is three halves okay which is one 1.5 so the y-intercept there is 0 1.5 .

Now to have our equation so x over a plus y over b is 1 so our equation will become x over 2 here our x intercept plus .

Y over y intercept 1.5 equals 1. so this is our final answer okay okay class let's try to solve these four problems if you wish to you can solve .

This on your own you can pause this video and try to check if makaparei okay now for number one okay let's put here the answers and here is the solution now for number one the midpoint of a .

Line segment pq is three four if abscissa of p is four and ordinate of q is negative two find the points p and q okay so i suggest let's draw first visualize so this is .

Pq line segment pq according here where the abscissa of p is for by the way class what is abscissa abscissa is the x coordinate okay x coordinate so .

Nothing memorized so for on the other hand cq given c y which is negative x sub one and we are given the midpoint which is three and four .

So for our solution for that now x bar so the formula for x bar is x one plus x2 over 2 where x sub 1 is 4 and x sub 2 is unknown so let's find x sub 2 over 2 which is equal to .

3 now transposing 2 on the other side believing 4 plus x plus 2 is x and x and x sub 2 is positive 2 so .

X sub 2 is positive okay let's erase this now let's solve for y sub 1 to solve for y sub 1 let's use the x y bar so y bar is just y sub 1 plus y sub two over two which is .

Y one plus negative two minus two over two is four rearranging is y1 minus 2 so y1 is just 8 plus 2 .

Which is 10 okay so see y1 is 10 okay so for answer for number one is b is four then .

On the other hand q is two negative two okay let's erase this now how about number two find a and b such that the midpoint of line segment pq is where p is a2 and q is one b is negative .

Three four okay for number two solution for two okay let's write the given so we have the midpoint which is negative three and four where p is .

A and two and q is 1b now let we are asked to find the value of a and b again let's just use the formula for the x bar and y bar since we are given the bitcoin .

So x bar so i will just straight to the point so x bar is just negative three given a plus one over two rearranging negative negative six is a plus one therefore a is negative six minus one .

Making negative seven so a is negative seven now y bar solving for y bar given that y bar is positive four which is two plus b over two .

Rearranging s two plus b then b is just eight minus two which is six so b is just positive six so the answer for number two is a is negative seven .

And b is positive six okay now let's erase this now for number three find the slope and y-intercept of y equals six .

So let us graph y equals six so solution so solution for number three sub into y equals six that's fine for the m and the y-intercept to graph y equals 6 so assuming this is positive y .

Positive x sabirito y is equal to 6 meaning the value of y is 6 regardless regard regardless what is the value of x so the graph of that the graph of y of y equals 6 is this so if this is 6 then this is the graph of y equals 6. .

Okay this is our line y equals six now take note this is a horizontal line okay what is the slope of a horizontal line the slope is automatically zero then the y-intercept okay the point where it touches the y-axis is .

Six which is six or it will be to be exact zero six okay okay so the answer for number three is m is zero y-intercept .

Six or zero six now number four we are down to our final problem find the equation of the line where the angle of inclination is pi over two red and passes negative five negative five meaning .

The angle between the horizontal axis and our line is pi over two let us convert first pi over two theta which is pi over two okay rad now let us convert rad into decrease first by multiplying 180 over pi so cancel cancel the theta is just 90 degrees .

So the angle of inclination is 90 degrees so technically if we will graph that so this is the angle of inclination 90 degrees and it passes negative 5 negative 5 so meaning our line is a .

Vertical line and it passes negative 5 negative 5 so if we will graph that so if this this is positive x negative x positive y negative y so assuming this is negative five .

Negative five so this is the point negative five negative five and we have a vertical line since the angle between the or the theta is 90 degrees so this is our line a standing line or a vertical line so this is the graph .

Now since this is a standing line this is a vertical line meaning the equation of this is just x equals negative 5 meaning the value of x is negative 5 regardless of the value of y so if the value of y is 10 still the value of x is negative 10. if .

The value is of 20 still the value of y is just negative five so that's why the equation of the line is just x equals negative five so the answer for number four is equation of the line is x equals negative five okay .

Okay okay this is our final answers now try to check if we if mag now sago and now to keep you motivated to study i will be giving twenty pesos worth of load or cash so first five democratic is assignment okay and perfect all or nothing .

And of course if naked is a video not then it's a discussion these problems are very easy okay easy peasy easy money okay number one find the value of k such that the distance from 3 negative k and 9 3 k is 5 k number 2 .

Find the distance from 1 to to the midpoint of the line segment whose endpoints are negative one negative three and negative three and negative five number three find the equation of the line given a angle of inclination is .

Five pi over six rad and ordinate of the y-intercept is negative four and what if the slope is negative three-fourths and passes through the midpoint of the line segment connecting negative four two and three negative three okay all you have to do is just .

Subscribe to my channel and send your complete solution to my facebook account kevin and belza namakiki tania screen okay thank you now before we end this discussion let us summarize first what we talk about .

At least my refreshing mama no first one okay two distinct points will determine a line so you have two points you connect them then you have a line okay this is a line this is array and this is a line segment okay take note the difference of the three .

Now second one the general equation of a line is a x plus b y plus c is zero okay this is called the linear equation it is because linear so the exponent is just is just one third the distance formula of the two points is .

D squared of x sub two minus x sub one quantity squared plus y sub two minus y sub 1 quantity squared okay number four the midpoint of the line segment which is denoted as capital m x bar y bar where the x bar is x sub 1 plus x sub 2 over 2 and y bar is .

Y sub 1 plus y sub 2 over 2. okay number five take note the angle of inclination is the angle above the horizontal axis and the angle of depression is the angle below the horizontal axis okay busta depression when you are feeling .

Depressed you are feeling low so back depression angle below okay how about this the slope of a line denoted by small m which is y sub 2 minus y sub 1 over x sub 2 minus x sub 1 or it could be if you are given theta which is the angle between the .

Horizontal axis and the line okay that is the data then we can use tangent theta opposite over adjacent which is just the rise of red okay okay and take note by the way class if marante on horizontal line the slope will automatically be .

Zero if vertical line then the slope is undefined take note of that okay and lastly let's talk about the equation of a line okay class by the way in choosing the equation of the line .

Choose what is best depends a given again depends a given if we are given a slope and an intercept then we use the slope intercept form y equals m x plus b where m is the slope b is the y intercept if we are given two points then we use this y minus y sub one .

Equals y sub two minus y sub one over x sub two minus x sub one which is just the slope times quantity x minus x sub one or if we are given point slope then we use this y minus y sub 1 is equal to m times x minus x sub 1 and lastly if we are given the .

Intercepts the x and y intercepts then we use the intercept form which is x over a which is the x-intercept plus y over b which is the y-intercept equals one again choose what is best dependence given okay and also don't forget to answer our .

Assignment to win 20 pesos worth of load or cash thank you see me on the part bye

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