<div style='float: left; width:100%;'> <h3 id="successconic-section-lecture---6success"><Success>Conic section lecture - 6</Success></h3> </div> <div style='float: right; width:1%;'> <!----> </div>

<h4 id="successpoints-of-intersections-of-the-line-with-the-parabola-success"><Success>Points of intersections of the line with the parabola </Success></h4> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p><strong>Points of intersections of the line $y=mx+c$ with the parabola $y^2=4ax$:</strong></p> <p>Three Possibilities:</p> <ol> <li>Two points of intersection.</li> <li>One points of intersection.</li> <li>No points of intersection.</li> </ol> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/20b6727f-613e-4be0-9d29-85fb835c7600/public"></p> </div>

<h4 id="successpoints-of-intersections-of-the-line-with-the-parabola-success-1"><Success>Points of intersections of the line with the parabola </Success></h4> <div style='float: left;text-align:left; width:100%;font-size:25px;'> <p>To find the points of intersection $y=mx+c$ and $y^2=4ax$, we solve</p> <p>$(mx+c)^2=4ax$</p> <p>$\Leftrightarrow m^2x^2 + 2mcx + c^2 = 4ax$</p> <p>$\Leftrightarrow m^2x^2 + 2(mc-2a)x + c^2 = 0$</p> <p>This is a quadratic equation in x, so it has either two real & distinct roots or equal roots or two non real complex roots.</p> <p>Disriminant = $[2(mc-2a)]^2 - 4m^2c^2$</p> <p>= $4 [m^2c^2 - 4amc + 4a^2 - m^2c^2]$</p> <p>= $16(a^2-amc)$</p> </div> <div style='float: right; width:0%;'> <!----> </div>

<h4 id="successpoints-of-intersections-of-the-line-with-the-parabola-success-2"><Success>Points of intersections of the line with the parabola </Success></h4> <div style='float: left;text-align:left; width:100%;font-size:25px;'> <p>$\therefore$ there are 2 points of intersection if $a^2-amc>0$</p> <p>that is $a>mc$</p> <p>There is one point of intersection if $a=mc$</p> <p>& No points if intersection if $a<mc$</p> <p>When $a=mc$ the line $y=mx+c$ intersects the parabola is only one point, and hence it is tangent to the parabola $y^2=4ax$ at the point (x,y) given by</p> <p>$x= \frac{-2(mc-2a)}{2m^2} = \frac{2(mc-2mc)}{2m^2}$</p> <p>$ = \frac{mc}{m^2} = \frac{c}{m}$</p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/91c55a04-8294-4ef0-1810-1d5f495a9c00/public"></p> </div>

<h3 id="successlength-of-chord-of-parabola-success"><Success>Length of chord of parabola </Success></h3> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p>Putting $x=\frac{c}{m}$ in the equation. $y=mx+c$ gives $y=m(\frac{c}{m}) + 2c$</p> <p>So, the line $y=mx+c$ is tangent to the parabola $y^2=4ax$ at the point ($\frac{c}{m},2c$) provided $mc=a$</p> <p><strong>Length of chord of parabola $y^2=4ax$:</strong></p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/b0451aa4-77a0-4f2d-61e6-e231c4a55b00/public"></p> </div>

<h3 id="successlength-of-chord-of-parabola-success-1"><Success>Length of chord of parabola </Success></h3> <div style='float: left;text-align:left; width:99%;font-size:30px;'> <p>Length of the chord joining ($x_1,y_2$) & ($x_2,y_2$) on the parabola $y^2=4ax$ is</p> <p>l = $\sqrt{(x_1 - x_2)^2 + (y_1 -y_2)^2}$</p> <p>Equation of the line joining ($x_1,x_2$) & ($x_2, y_2$) is</p> <p>$y-y_1 = \frac{y_1-y_2}{x_1-x_2} (x-x_1)$</p> <p>$y=mx+c$, where $m = \frac{y_1-y_2}{x_1-x_2}$</p> <p>$c=y_1 - (\frac{y_1-y_2}{x_1-x_2}) x_1$</p> </div> ====== <h3 id="successlength-of-chord-of-parabola-success-2"><Success>Length of chord of parabola </Success></h3> <div style='float: left;text-align:left; width:99%;font-size:30px;'> <p>Provided $x_1 \neq x_2$</p> <p>Note that if $x_1=x_2$ then</p> <p>$l=|y_1-y_2|$</p> <p>$y_1 ^2 = 4ax , y_2 ^2 = 4ax$</p> <p>$y_1=-y_2$ & $|y_1-y_2|=2\sqrt{4ax_1} = 4\sqrt{ax_1}$</p> <p>$x_1$ and $x_2$ satisfy the quadratic equation.</p> </div> <div style='float: right; width:1%;'> <!----> </div>

<h3 id="successlength-of-chord-of-parabola-success-3"><Success>Length of chord of parabola </Success></h3> <div style='float: left;text-align:left; width:99%;font-size:30px;'> <p>$m^2x^2 + 2(mc-2a)x + c^2= 0$</p> <p>$\therefore x_1 + x_2 = \frac{-2(mc-2a)} { m^2}$</p> <p>& $x_1 x_2 = \frac{c^2}{m^2}$</p> <p>$\therefore (x_1-x_2)^2 = (x_1 + x_2)^2 - 4x_1x_2$</p> <p>$ = \frac{4}{m^4} (mc-2a)^2 - \frac{4c^2}{m^2}$</p> <p>$ = \frac{4}{m^4} [m^2 c^2 - 4amc + 4a^2 - m^2c^2]$</p> <p>$ = \frac {16 a (a-mc)}{m^4}$</p> <p>Also, $y_1-y_2 = (mx_1 + c) - (mx_2 + c) = m(x_1 - x_2)$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/730f9da4-7f3f-4a9b-cd76-26ef757b4200/public"></p> </div>

<h3 id="successlength-of-chord-of-parabola-success-4"><Success>Length of chord of parabola </Success></h3> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$\therefore$ Length of the chord, l = $\sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}$</p> <p>= $\sqrt{(x_1 -x_2)^2 + m^2 (x_1 - x_2)^2}$</p> <p>= $\sqrt{1+m^2} | x_1-x_2 |$</p> <p>= $\sqrt{1+m^2} \frac{4}{m^2} \sqrt{a (a-mc)}$</p> <p>To get the formula for l in terms of ($x_1, x_2$) & ($y_1,y_2$) we can put</p> <p>$m = \frac{y_1-y_2}{x_1-x_2}$ & $c= y_1 - (\frac{y_1-y_2}{x_1-x_2}) x_1$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/a5e7246b-1744-4653-f707-d904bc6e7b00/public"></p> </div>

<h4 id="successequation-of-the-tangent-line-to-the-parabolasuccess"><Success>Equation of the tangent line to the parabola</Success></h4> <div style='float: left;text-align:left; width:50%;font-size:25px;'> <p><strong>Equation of the tangent line to the parabola $y^2=4ax$ at a point ($x_1,x_2$):</strong></p> <p>Suppose the line has slope m then the equation of the tangent line is given by $y-y_1=m(x-x_1)$</p> <p>i.e $y=mx+(y_1-mx_1)$</p> <p>We know that this line is tangent to $y^2=4ax$</p> <p>if $a=mc = m(y_1-mx_1)$</p> <p>$\Leftrightarrow m^2 x_1 - my_1 + a =0$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c45c527a-f054-4bf3-cd37-7c7beb0c9d00/public"></p> </div>

<h4 id="successequation-of-the-tangent-line-to-the-parabolasuccess-1"><Success>Equation of the tangent line to the parabola</Success></h4> <div style='float: left;text-align:left; width:100%;font-size:25px;'> <p>$\Leftrightarrow m = \frac{y_1 \pm \sqrt{y_1 ^2 - 4ax_1}}{2x_1}$, But $y_1^2 = 4ax_1$</p> <p>So, $m = \frac{y_1}{2x_1}$ This can also be written as</p> <p>$m = \frac{y_1 ^2}{ax_1y_1} = \frac{4ax_1}{ax_1y_1} = \frac{2a}{y_1}$</p> <p>$m = \frac{2a}{y_1}$</p> <p>Equation of the tangent line is</p> <p>$y=mx+(y_1-mx_1)$</p> <p>= $\frac{2a}{y_1} x +y_1 - \frac{y_1}{2}$</p> <p>= $\frac{2ax}{y_1} + \frac{y_1}{2}$</p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c2cc642c-20eb-4311-08ba-877a09498e00/public"></p> </div>

<h4 id="successequation-of-the-tangent-line-to-the-parabolasuccess-2"><Success>Equation of the tangent line to the parabola</Success></h4> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$\Leftrightarrow yy_1 = 2ax + \frac{y^2}{2} = 2ax + \frac{4ax_1}{2}$</p> <p>$\Leftrightarrow yy_1 = 2a(x+x_1)$</p> <p>Equation of the tangent line to the parabola $y^2=4ax$ at the point ($x_1,y_1$) on the parabola.</p> </div> <div style='float: right; width:1%;'> <!----> </div>

<h4 id="successderivation-of-the-equation-of-the-tangent-line-using-calculassuccess"><Success>Derivation of the equation of the tangent line using calculas</Success></h4> <div style='float: left;text-align:left; width:99%;font-size:30px;'> <p>$y^2=4ax$: Equation of the parabola ($x_1,y_1$) lies on the parabola $\Rightarrow y_1^2 = 4ax_1$. we know that the slope of the tangent line at a point ($x_1,y_1$) on any curve $y=f(x)$ is given by $m=\frac{dy}{dx} |_{e(x_1,y_1)}$</p> <p>$y^2=4ax \Rightarrow 2y \frac{dy}{dx} = 4a$</p> <p>$\Rightarrow \frac{dy}{dx} = \frac{2a}{y} \Rightarrow m=\frac{2a}{y_1}$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e09dc31d-dbd5-4aa1-b9ed-6d470948e300/public"></p> </div>

<h4 id="successderivation-of-the-equation-of-the-tangent-line-using-calculassuccess-1"><Success>Derivation of the equation of the tangent line using calculas</Success></h4> <div style='float: left;text-align:left; width:99%;font-size:25px;'> <p>$\therefore$ equation of the tangent line is</p> <p>$y-y_1 = \frac{2a}{y_1} (x-x_1)$</p> <p>$\Leftrightarrow yy_1 - y_1 ^2 = 2ax-2ax_1$</p> <p>But $y_1 ^2 = 4ax_1$</p> <p>$\therefore yy_1 - 4ax_1 = 2ax - 2ax_1$</p> <p>$\Leftrightarrow yy_1 = 2ax + 2ax_1 $</p> <p>$\Leftrightarrow yy_1 = 2a(x+x_1)$</p> <p><strong>Remark:</strong> The above derivation assumed</p> <p>$(x_1,y_1)\neq 0$</p> </div> <div style='float: right; width:1%;'> <!----> </div>

<h4 id="successderivation-of-the-equation-of-the-tangent-line-using-calculassuccess-2"><Success>Derivation of the equation of the tangent line using calculas</Success></h4> <div style='float: left;text-align:left; width:50%;font-size:25px;'> <p>However, if the point $(x_1,y_1)=(0,0)$ Then it is clear that the tangent line at (0,0) is the y-axis, where equation is x=0.</p> <p>If we put $(x_1,y_1)=(0,0)$ in the equation obtained $yy_1 = 2a (x+x_1)$</p> <p>we get 0 = 2a(x+0) i.e x=0</p> <p>Hence, the equation $yy_1 = 2a (x+x_1)$ is valid for even the point $(0,0)$</p> <p>general equation of tangent at $(x_1,y_1)$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/593edeb2-724d-4638-c658-66d793cf1900/public"></p> </div>

<h4 id="successequation-of-the-normal-at-point-at-the-parabolasuccess"><Success>Equation of the normal at point at the parabola</Success></h4> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p><strong>Equation of the normal at point $(x_1,y_1)$ on the parabola $y^2=4ax$:</strong></p> <p>Equation of the tangent is $yy_1=2a(x+x_1)$</p> <p>If $y_1 \neq 0$ then $y=\frac{2a}{y_1} (x+x_1)$</p> <p>$\therefore$ Slope of tangent = $\frac{2a}{y_1}$</p> <p>$\Rightarrow$ Slope of normal, m = $\frac{-y_1}{2a}$</p> <p>$\therefore$ Equation of normal is</p> <p>$y-y_1 = \frac{-y_1}{2a} (x-x_1)$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/72f5dd7c-9e4a-4684-8952-ea5ee8873b00/public"></p> </div>

<h4 id="successequation-of-the-normal-at-point-at-the-parabolasuccess-1"><Success>Equation of the normal at point at the parabola</Success></h4> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p><strong>In terms of the slope m</strong>:</p> <p>$m=\frac{-y_1}{2a}$ i.e $y_1 = -2am$</p> <p>$\therefore x_1 = \frac{y_1 ^2} {4a} = \frac{4a^2 m^2 }{4a} = am^2$</p> <p>Putting in the equation. $y-y_1 = \frac{-y_1}{2a} (x-x_1)$,</p> <p>we get</p> <p>$y+2am=m(x-am^2)$</p> <p>$\Leftrightarrow y=mx-2am-am^3$</p> <p>equation of the normal at the points $(x_1,y_1) = (am^2,-2am)$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/53dc87af-40ad-433e-8f35-ceda760b2900/public"></p> </div>

<h3 id="successsubtanget-and-subnormalsuccess"><Success>Subtanget and Subnormal</Success></h3> <div style='float: left;text-align:left; width:50%;font-size:25px;'> <p>AT: subtangent</p> <p>AN: subnormal</p> <p>Equation of line PT is</p> <p>$yy_1 = 2a(x+x_1)$</p> <p>Putting $y=0$ gives $x=-x_1$</p> <p>$\therefore T \equiv (-x_1,0)$</p> <p>Subtangent $AT = 2x_1$</p> <p>$\triangle TAP$ is similar to $\triangle PAN$</p> <p>$\Rightarrow \frac{AN}{AP} = \frac{AP}{AT} \Rightarrow AN = \frac{AP^2}{AT} = \frac{y_1^2}{2x_1} = \frac{4ax_1}{2x_1} = 2a$</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/68431a4c-ba6a-41a9-94aa-e33ce7383b00/public"></p> </div>

<h3 id="successsubtanget-and-subnormalsuccess-1"><Success>Subtanget and Subnormal</Success></h3> <!----> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$\therefore$ Subnormal, $AN = 2a$</p> <p>Which is a constant</p> </div> <div style='float: right; width:1%;'> <!----> </div>

<div> <h3 id="successthank-yousuccess"><Success>Thank you</Success></h3> </div>