WEBVTT
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the question they are asking which of the following four
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? Lines are parallel and are identical. So the
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Lines are given by the equations L one X equal
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to one place 60, Why Equal to one minus
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treaty? And they recorded 12 3 plus five.
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And I'll do is given by the equation X equal
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to one plus two, T Y equal to T
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Z is equal to one plus 40. And Elstree
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is given by the equation two x minus two equal
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to four minus four, Y is equal to that
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plus one is equal to T. C. and
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L four is given by the equation are equal to
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315 Plus t. 4-8 in victor format. So
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in order to find the solution of the given question
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first we have to find the equation of each of
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the lines in regular forms. So and one line
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has a normal victor equal to six minus three,
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12 and a point passing through it is equal to
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115. Therefore the equation of the line is equal
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to six into X-1-3, 2, Y
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-1 Plus 12 into that-5 is equal to zero
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. And after solving this equation though, Equation for
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L one Is equal to two,-Y plus or
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that is equal to 21. And similarly the Equation
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for L two can be written in the regular form
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as L two has the Normal vector is equal to
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214 and the point that passes through this line is
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equal to 101 and therefore the equation of the Line
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, L two is equal to two into X-1
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Plus one into Y-0 0 plus four into Z
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-1 equal to zero. After solving this equation,
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no final equation of L two is two, X
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plus y plus four. Zed is equal to six
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and the aggression for Elstree can written as X equal
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to t plus two by two, Y equal to
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four minus T by four and That is equal to
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T-1. Therefore the normal vector of this line
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is equal to 0.5-1 x four and 1 as
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this is infraction form. So we can multiply this
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vector Bayous killer 1 24 so this is equal to
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two-1 and four. Therefore from the above equation
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we find up Point that passes through this line is
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equal to 1,-1. Therefore the equation of
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this line is equal to two into x minus one
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minus one into y minus one plus four in two
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, Z plus one is equal to zero. So
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the final equation for l C is equal to two
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, x minus y plus four, Z is equal
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to minus three. And The question for line L
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four is did I from the Normal victor that is
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equal to four 28. And the Point that passes
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through this line that is equal to 315. Therefore
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the equation of this line can be written as four
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into x minus three plus two into y minus one
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plus eight into zero minus five is equal to zero
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. And the final equation that can region for the
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line, L four is equal to two, x
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plus y plus or that is equal to 27.
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Mhm. So the equations of the line, L
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one, L 2, L three and L four
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. As we have already found out using the values
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of the normal vector and the coordinates from each of
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the equations that was given in the question to be
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written in the regular form. And so in order
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to find radio, these lines are parallel or identical
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. Mhm. Thank you. We have to check
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For the first pair of lines that is L one
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and L two. Yeah that is a condition is
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whether there are parallel or not. So in order
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to find their parallel or not to compare the coefficients
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of the X, y and z coordinates of these
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two lines. So this is equal to two by
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two is equal to minus one by one is equal
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to four by four. After simplifying these ratios,
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we get this equal to one not equal to-1
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, not equal to one. Therefore L one and
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L two R not parallel since they are not parallel
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, they are not identical. Also in the second
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condition we have to take for the next pair of
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lines that is that is L one and L three
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condition, he is whether there are parallel or not
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. So in order to find their parallel or not
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we have to compare the coefficients of each other coordinates
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of these two lines. So the issues are two
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x 2 is equal to-1 x-1, equal
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to four x 4. So after simplifying these ratios
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, we get It to be equal to one,
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That is equal to one equal to one. Therefore
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L one and else we are Palin hence we have
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to check whether these are identical or not to check
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it, we have to Put the value of x
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equal to y equal to zero in border equations of
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the line L one and else trees. So the
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first line L one here, the question would be
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that equal to 21 x four. And so therefore
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the point or The X, YZ coordinates would be
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0, 0 and 21 x four. And therefore
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For the line three Putting the same conditions, that
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is equal to-3 x four and therefore your point
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is 00 and-3 x four. Therefore, since
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the coordinates are not equal, so L one and
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L three are not identical. Yeah. Yeah.
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And for the third case for the third line pair
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, that is L one and L four, we
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have to take the same conditions first, is there
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parallel or not? So comparing the coefficients, the
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coordinates of these two lines, we get the ratio
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is two by two equal to minus one by one
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equal to four by four. So after simplifying these
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ratios we get It will be equal to one is
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not equal to-1 Is not equal to one.
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Therefore L one and L four are not Berlin.
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Therefore these are not identical also to take next pair
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of lines that is L two and L three.
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The first condition to check whether these are valid or
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not. So in order to check we have to
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find the ratio between the coefficients of each other coordinates
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of these two lines. Therefore the Ratios are two
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x 2 is equal to one x-1 equal to
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four x 4. So after simplifying these ratios we
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get the value to be equal to one, not
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equal to minus one, not equal to one.
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Therefore L two and L three are not parallel and
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hence these are not identical. Also the next spirit
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of lying is and To an L four. Therefore
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taking the first condition that is whether these lines are
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parallel or not. So comparing the coefficients of the
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coordinates of these two lines, the ratio becomes two
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by two is equal to one by one equal to
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four by four, softer, simply paying these ratios
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speak get to be equal to one equal to one
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equal to one. Therefore L two and L four
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are parallel lines and hence we have to check the
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second condition in the question that is whether they're identical
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or not. So You put the value of x
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equal to y equal to zero To find the value
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of that and all the coordinates each of the lines
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. So for the line L- two that is equal
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to six by four. And Therefore the point for
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this is equal to 00 and six x 4.
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And the point for the line L four is XY
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equal to zero, so that is equal to 27
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x four. So here the point is equal to
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00, x four. Hence since.1 is not
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equal to point to, therefore L two and L
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four are not identical. Mhm I'm checking the last
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pair of lines that is L three and and four
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. So the yeah, first condition to check whether
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these lines are parallel or not. So the yeah
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condition to check this is we have to compare the
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coefficients of all the coordinates of these lines. So
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the issues are two by two Is equal to-1
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x one equal to four x 4. Therefore,
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after simplifying the ratios we get to be equal to
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one is not equal to-1 is not equal to
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one. Therefore L three and L four, R
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not parallel. Hence these lines are also not identical
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. So from the above conclusions, we find the
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answer for this question to be L two and L
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four are parallel lines but not identical and L one
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and L three are also palette lines, I'm not
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identical, and this is the required answer to the
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given question.