Length of normal. chord…
$begingroup$
What is the length of normal chord which subtends right angle at the vertex of parabola $y^2=4x$. $$My Try$$ let the equation of normal be $y=mx-am^3-2am$ Now I assumed slope of this line as $45$ (which is probably wrong). So I got $y=x-3$ thus solving for $y^2=4x$ and the calculated equation I got $x=9,y=6$ ignoring the smaller root. As answers given are more in magnitude. So I got length from focus $1,0$ as $10$ but it is wrong. Thanks!
analytic-geometry conic-sections
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
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add a comment |
$begingroup$
What is the length of normal chord which subtends right angle at the vertex of parabola $y^2=4x$. $$My Try$$ let the equation of normal be $y=mx-am^3-2am$ Now I assumed slope of this line as $45$ (which is probably wrong). So I got $y=x-3$ thus solving for $y^2=4x$ and the calculated equation I got $x=9,y=6$ ignoring the smaller root. As answers given are more in magnitude. So I got length from focus $1,0$ as $10$ but it is wrong. Thanks!
analytic-geometry conic-sections
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
$endgroup$
add a comment |
$begingroup$
What is the length of normal chord which subtends right angle at the vertex of parabola $y^2=4x$. $$My Try$$ let the equation of normal be $y=mx-am^3-2am$ Now I assumed slope of this line as $45$ (which is probably wrong). So I got $y=x-3$ thus solving for $y^2=4x$ and the calculated equation I got $x=9,y=6$ ignoring the smaller root. As answers given are more in magnitude. So I got length from focus $1,0$ as $10$ but it is wrong. Thanks!
analytic-geometry conic-sections
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
$endgroup$
What is the length of normal chord which subtends right angle at the vertex of parabola $y^2=4x$. $$My Try$$ let the equation of normal be $y=mx-am^3-2am$ Now I assumed slope of this line as $45$ (which is probably wrong). So I got $y=x-3$ thus solving for $y^2=4x$ and the calculated equation I got $x=9,y=6$ ignoring the smaller root. As answers given are more in magnitude. So I got length from focus $1,0$ as $10$ but it is wrong. Thanks!
analytic-geometry conic-sections
analytic-geometry conic-sections
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
asked Jun 9 '16 at 15:53
Archis WelankarArchis Welankar
12k41642
12k41642
add a comment |
add a comment |
1 Answer
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"Subtends right angle"? Do you mean a line segment AB such that A and B lie on the parabola and angle AOB is a right angle? IF the A and B are $(x, 2sqrt{x})$ and $(x,-2sqrt{x})$ Then the length of the line segment is, of course, just the difference in y values, $4sqrt{x}$. That would be the case when the two sides of the angle make 45 degree angles with the x-axis.
But that is not the only possible case. If, say, the upper side makes angle $theta$ with the x-axis, it can be written $y= tan(theta)x$ and crosses the parabla where $tan(theta)x= 2sqrt{x}$ so $sqrt{x}= 2cot(theta)$, $x= 4 cot^2(theta)$, $y= 2sqrt{x}= 2(2 cot(theta)= 4 cot(theta)$. The other line, which is perpendicular to the first, so has slope $-frac{1}{tan(theta)}= -cot(theta)$, has equation $y= -cot(theta)x$ crosses the parabola where $-cot(theta)x= -2sqrt{x}$ so $sqrt{x}= -2tan(theta)$, $x= 4 tan^2(theta)$, $y= -4 tan(theta)$. Can you find the distance between points $A= (4 cot^2(theta), 4 cot(theta))$ and $B= (4 tan(theta), -4tan(theta))$?
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
$endgroup$
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
add a comment |
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$begingroup$
"Subtends right angle"? Do you mean a line segment AB such that A and B lie on the parabola and angle AOB is a right angle? IF the A and B are $(x, 2sqrt{x})$ and $(x,-2sqrt{x})$ Then the length of the line segment is, of course, just the difference in y values, $4sqrt{x}$. That would be the case when the two sides of the angle make 45 degree angles with the x-axis.
But that is not the only possible case. If, say, the upper side makes angle $theta$ with the x-axis, it can be written $y= tan(theta)x$ and crosses the parabla where $tan(theta)x= 2sqrt{x}$ so $sqrt{x}= 2cot(theta)$, $x= 4 cot^2(theta)$, $y= 2sqrt{x}= 2(2 cot(theta)= 4 cot(theta)$. The other line, which is perpendicular to the first, so has slope $-frac{1}{tan(theta)}= -cot(theta)$, has equation $y= -cot(theta)x$ crosses the parabola where $-cot(theta)x= -2sqrt{x}$ so $sqrt{x}= -2tan(theta)$, $x= 4 tan^2(theta)$, $y= -4 tan(theta)$. Can you find the distance between points $A= (4 cot^2(theta), 4 cot(theta))$ and $B= (4 tan(theta), -4tan(theta))$?
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
$endgroup$
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
add a comment |
$begingroup$
"Subtends right angle"? Do you mean a line segment AB such that A and B lie on the parabola and angle AOB is a right angle? IF the A and B are $(x, 2sqrt{x})$ and $(x,-2sqrt{x})$ Then the length of the line segment is, of course, just the difference in y values, $4sqrt{x}$. That would be the case when the two sides of the angle make 45 degree angles with the x-axis.
But that is not the only possible case. If, say, the upper side makes angle $theta$ with the x-axis, it can be written $y= tan(theta)x$ and crosses the parabla where $tan(theta)x= 2sqrt{x}$ so $sqrt{x}= 2cot(theta)$, $x= 4 cot^2(theta)$, $y= 2sqrt{x}= 2(2 cot(theta)= 4 cot(theta)$. The other line, which is perpendicular to the first, so has slope $-frac{1}{tan(theta)}= -cot(theta)$, has equation $y= -cot(theta)x$ crosses the parabola where $-cot(theta)x= -2sqrt{x}$ so $sqrt{x}= -2tan(theta)$, $x= 4 tan^2(theta)$, $y= -4 tan(theta)$. Can you find the distance between points $A= (4 cot^2(theta), 4 cot(theta))$ and $B= (4 tan(theta), -4tan(theta))$?
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
$endgroup$
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
add a comment |
$begingroup$
"Subtends right angle"? Do you mean a line segment AB such that A and B lie on the parabola and angle AOB is a right angle? IF the A and B are $(x, 2sqrt{x})$ and $(x,-2sqrt{x})$ Then the length of the line segment is, of course, just the difference in y values, $4sqrt{x}$. That would be the case when the two sides of the angle make 45 degree angles with the x-axis.
But that is not the only possible case. If, say, the upper side makes angle $theta$ with the x-axis, it can be written $y= tan(theta)x$ and crosses the parabla where $tan(theta)x= 2sqrt{x}$ so $sqrt{x}= 2cot(theta)$, $x= 4 cot^2(theta)$, $y= 2sqrt{x}= 2(2 cot(theta)= 4 cot(theta)$. The other line, which is perpendicular to the first, so has slope $-frac{1}{tan(theta)}= -cot(theta)$, has equation $y= -cot(theta)x$ crosses the parabola where $-cot(theta)x= -2sqrt{x}$ so $sqrt{x}= -2tan(theta)$, $x= 4 tan^2(theta)$, $y= -4 tan(theta)$. Can you find the distance between points $A= (4 cot^2(theta), 4 cot(theta))$ and $B= (4 tan(theta), -4tan(theta))$?
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
$endgroup$
"Subtends right angle"? Do you mean a line segment AB such that A and B lie on the parabola and angle AOB is a right angle? IF the A and B are $(x, 2sqrt{x})$ and $(x,-2sqrt{x})$ Then the length of the line segment is, of course, just the difference in y values, $4sqrt{x}$. That would be the case when the two sides of the angle make 45 degree angles with the x-axis.
But that is not the only possible case. If, say, the upper side makes angle $theta$ with the x-axis, it can be written $y= tan(theta)x$ and crosses the parabla where $tan(theta)x= 2sqrt{x}$ so $sqrt{x}= 2cot(theta)$, $x= 4 cot^2(theta)$, $y= 2sqrt{x}= 2(2 cot(theta)= 4 cot(theta)$. The other line, which is perpendicular to the first, so has slope $-frac{1}{tan(theta)}= -cot(theta)$, has equation $y= -cot(theta)x$ crosses the parabola where $-cot(theta)x= -2sqrt{x}$ so $sqrt{x}= -2tan(theta)$, $x= 4 tan^2(theta)$, $y= -4 tan(theta)$. Can you find the distance between points $A= (4 cot^2(theta), 4 cot(theta))$ and $B= (4 tan(theta), -4tan(theta))$?
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
answered Jun 9 '16 at 16:33
user247327user247327
11.1k1515
11.1k1515
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
add a comment |
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
Ya so how would you calculate the length
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:39
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
$begingroup$
I have done some same manipulations but all in vain can you cast some more shadow on it
$endgroup$
– Archis Welankar
Jun 9 '16 at 16:41
add a comment |
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