mathematics-class-11-unit-11-chapter-2-straight-lines-l-2-6-wncn38qokzg

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example 1</B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:100%;font-size:22px;'>
<p>Show that the line joining $(2,-3)$ and $(-5,1)$ is</p>
<p><strong>(i)</strong> parallel to the line joining $(7,-1)$ and $(0,3)$</p>
<p><strong>(ii)</strong> perpendicular to the line joining $(4,5)$ and $(0,-2)$</p>
</div>
</main>
<hr>
<footer style='font-size:18px'>
Parallel and perpendicular lines$\to$<span style="color:blue">Example 1</span>$\to$Solution$\to$Angle between two lines$\to$Example 2$\to$Example 3$\to$Equation of a straight line$\to$Equation of a line parallel to x-axis$\to$Equation of a line parallel to y-axis$\to$Example 4$\to$Equation of straight line in various standard forms
</footer>
<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Solution</B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:100%;font-size:22px;'>
<p>$\quad P(2,-3) \text{and} Q(-5,1)$</p>
<p>$\therefore \text{slope of} \ P Q =\frac{1+3}{-5-2}=\frac{4}{-7}$</p>
<p><strong>(i)</strong> $A(7,-1)$ and $B(0,3)$</p>
<p>$\therefore$ slope of $A B=\frac{3+1}{0-7}=\frac{4}{-7}$</p>
<p>$\therefore$ slope of $P Q=$ slope of $A B=-\frac{4}{7}$</p>
<p>$\therefore \quad P Q || A B$</p>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-11-chapter-2-straight-lines-l-2_6-wncn38qokzg/img/000001.png"></p>
</div>
</main>
<hr>
<footer style='font-size:18px'>
Parallel and perpendicular lines$\to$Example 1$\to$<span style="color:blue">Solution</span>$\to$Angle between two lines$\to$Example 2$\to$Example 3$\to$Equation of a straight line$\to$Equation of a line parallel to x-axis$\to$Equation of a line parallel to y-axis$\to$Example 4$\to$Equation of straight line in various standard forms
</footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Example 4</B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:30%;font-size:22px;'>
<p>Find the equation of the line passing through $(2,3)$ and is</p>
<p><strong>(i)</strong> parallel to $x$-axis</p>
<p><strong>(ii)</strong> parallel to $y$-axis.</p>
</div>
<div style='float: left; text-align:left; width:70%;font-size:22px;'>
<div style='float: right; width:50%;'>
<img src ="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/cbcbe429-5723-4420-8658-958c316ff200/public" widht ="50%">
</div>
<div style='float: right; width:50%;'>
<img src ="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5d359be0-4216-4618-249b-aa4308945e00/public" widht ="50%">
</div>
</main>
<hr>
<footer style='font-size:18px'>
Example 3$\to$Solution$\to$Equation of a straight line$\to$Equation of a line parallel to x-axis$\to$Equation of a line parallel to y-axis$\to$<span style="color:blue">Example 4</span>$\to$Equation of straight line in various standard forms$\to$Equation of straight line: slope-intercept form$\to$Slope - intercept form
</footer>
<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B><strong>Solution</strong></B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:50%;font-size:22px;'>
<p>$y =3$</p>
<p>$\therefore \quad$ Equation of $l_1$ is y = 3</p>
<p>$x=2$</p>
<p>Equation of $l_2$ is $x=2$</p>
</div>
<div style='float: right; width:40%;'>
<img src ="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/96d562ed-fb3c-4ee3-6670-325f80fe3f00/public" width ="70%">

</div>
</main>
<hr>
<footer style='font-size:18px'>
Example 3$\to$<span style="color:blue">Solution</span>$\to$Equation of a straight line$\to$Equation of a line parallel to x-axis$\to$Equation of a line parallel to y-axis$\to$Example 4$\to$Equation of straight line in various standard forms$\to$Equation of straight line: slope-intercept form$\to$Slope - intercept form
</footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Equation of straight line in various standard forms</B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:60%;font-size:22px;'>
<p><strong>1. Point- Slope Form:</strong></p>
<p>Let line  passing through $P(x_1,y_1)$</p>
<p>with slope $m$.</p>
<p>$m=\tan \theta=\frac{Q R}{P R}=\frac{y-y_1}{x-x_1}$</p>
<p>$\Rightarrow(y-y_1)=m(x-x_1)$</p>
</div>
<div style='float: right; width:40%;'>
<p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/77fea928-2c8a-4a3c-9305-606968bb4300/public"></p>

</div>
</main>
<hr>
<footer style='font-size:18px'>
Example 3$\to$Solution$\to$Equation of a straight line$\to$Equation of a line parallel to x-axis$\to$Equation of a line parallel to y-axis$\to$Example 4$\to$<span style="color:blue">Equation of straight line in various standard forms</span>$\to$Equation of straight line: slope-intercept form$\to$Slope - intercept form
</footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Equation of straight line: perpendicular or normal form </B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:60%;font-size:22px;'>
<p>$A(p \cos \alpha, p \sin \alpha)$</p>
<p>$\therefore \quad$ slope of OA $=\tan \alpha$ Since OA is perpendicular to $l$ (given)</p>
<p>$\Rightarrow$ slope of $l$ ie. $m=-\frac{1}{\tan \alpha} = -\cot \alpha$</p>
<p>Equation of $l$ passing through</p>
<p>$A(p \cos \alpha, p \sin \alpha)\text {with slope}$</p>
<p>$m=-\cot \alpha$ is</p>
</div>
<div style='float: right; width:30%;'>
<p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/b434f0bc-fff7-46ef-0e29-c7156c460000/public"></p>

</div>
</main>
<hr>
<footer style='font-size:18px'>
Equation of straight line: slope-intercept form$\to$Slope - intercept form$\to$Equation of straight line: two point form$\to$Equation of straight line: intercept form$\to$<span style="color:blue">Equation of straight line: perpendicular or normal form</span>$\to$Equation of straight line: general form
</footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Equation of straight line: perpendicular or normal form </B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:100%;font-size:22px;'>
<p>$(y-y_1)=m(x-x_1)$</p>
<p>$\Rightarrow  (y-p \sin \alpha)=m(x-p \cos \alpha)$</p>
<p>$\Rightarrow (y-p \sin \alpha) =-\frac{\cos \alpha}{-\sin \alpha}(x-p \cos \alpha) $</p>
<p>$\Rightarrow  y \sin \alpha-p \sin ^2 \alpha=-x \cos\alpha+p \cos ^2 \alpha$</p>
<p>$\Rightarrow x \cos \alpha+y \sin \alpha =p \sin ^2 \alpha+p \cos ^2 \alpha $</p>
<p>$\quad \quad =p(\sin ^2 \alpha+\cos ^2 \alpha) =p $</p>
<p>$x \cos \alpha+y \sin \alpha  =p$</p>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-11-chapter-2-straight-lines-l-2_6-wncn38qokzg/img/386-2998.8.jpg"></p>
</div>
</main>
<hr>
<footer style='font-size:18px'>
Equation of straight line: slope-intercept form$\to$Slope - intercept form$\to$Equation of straight line: two point form$\to$Equation of straight line: intercept form$\to$<span style="color:blue">Equation of straight line: perpendicular or normal form</span>$\to$Equation of straight line: general form
</footer>

<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Equation of straight line: general form </B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:100%;font-size:22px;'>
<p>$A x+B y+C=0, \quad$$A,B$ and $C \in R .$</p>
<p>$A$ and $B$ not both equal to zero.</p>
<ol>
<li>
<p>${A=0} \Rightarrow B y+C=0 \Rightarrow y=-C / B$</p>
</li>
<li>
<p>$B=0 \Rightarrow A x+C=0 \Rightarrow x=-C / A$ and
$A \neq 0$</p>
</li>
<li>
<p>$A \neq 0$ and $B \neq 0$</p>
</li>
</ol>
</div>  
</main>
<hr>
<footer style='font-size:18px'>
Equation of straight line: slope-intercept form$\to$Slope - intercept form$\to$Equation of straight line: two point form$\to$Equation of straight line: intercept form$\to$Equation of straight line: perpendicular or normal form$\to$<span style="color:blue">Equation of straight line: general form</span>
</footer>
<p>======</p>
<p><Success style="float:center;text-align:center;font-size:28px;" ><B>Straight line lecture 2</B></Success></p>
<p><Primary style="float:center;text-align:center;font-size:24px;"><B>Equation of straight line: general form </B></Primary></p>
<hr>
<main>
<div style='float: left; text-align:left; width:100%;font-size:22px;'>
<p>$ A x+B y+c=0$</p>
<p>$ \Rightarrow \quad B y=-A x-C $</p>
<p>$\Rightarrow \quad y=-\frac{A}{B} x-\frac{C}{B}$</p>
</div>
<div style='float: right; width:0%;'>
<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-11-chapter-2-straight-lines-l-2_6-wncn38qokzg/img/423-3268.8.jpg"></p>
</div>
</main>
<hr>
<footer style='font-size:18px'>
Equation of straight line: slope-intercept form$\to$Slope - intercept form$\to$Equation of straight line: two point form$\to$Equation of straight line: intercept form$\to$Equation of straight line: perpendicular or normal form$\to$<span style="color:blue">Equation of straight line: general form</span>
</footer>