Find the coordinates of intersection between tangents and given curve

In summary, the problem involves finding the equations of two straight lines that are tangent to the curve ##y=4-x^2## and pass through the point ##(-1,7)##. This can be done by using the equation of a tangent line and substituting the given point to solve for the slope, and then using the slope to solve for the equations of the lines. Alternatively, one can use the derivative of the curve at the given point to find the slope, and then proceed with finding the equations of the lines.
  • #1
chwala
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Homework Statement
Find the co-ordinates of intersection between tangents that pass through the points ##(-1,7)## and the curve ##y=4-x^2##.
Relevant Equations
differentiation
ooops...this was a bit tricky but anyway my approach;

...
##\dfrac{dy}{dx}=-2x##

therefore;

##\dfrac{y-7}{x+1}=-2x##

and given that, ##y=4-x^2## then;

##4-x^2-7=-2x^2-2x##

##x^2+2x-3=0##

it follows that, ##(x_1,y_1)=(-3,-5)## and ##(x_2,y_2)=(1,3)##.

There may be another approach, your insight is welcome guys!.
 
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  • #2
tangents that pass through the points (1,−7)
Do they ?

##\ ##
 
  • #3
BvU said:
Do they ?

##\ ##
just amended question...was a typo ...supposed to be ##(-1,7)##
 
  • #4
:mad:
 
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  • #5
The way I understand this is: there are two lines tangent to ##y:=4-x^2## that pass through ##(-1,7)##. The task is to find, where these lines intersect with ##y##.

So the tangent is of the form ##z-7 = k(x+1)##, where ##k## is open. We have ##z=y## such that there is exactly one point of intersection (because we want it to be tangent). So ##4-x^2 = kx+ k+7##, i.e, ##x^2 +kx + (k+3) = 0##. This yields the criterion ##k^2 - 4(k+3)=0## for ##k##.
 
  • #6
BvU said:
:mad:
Aaargh @BvU Long time man! Hope you good...cheers man.
 
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  • #7
nuuskur said:
The way I understand this is: there are two lines tangent to ##y:=4-x^2## that pass through ##(-1,7)##. The task is to find, where these lines intersect with ##y##.

So the tangent is of the form ##z-7 = k(x+1)##, where ##k## is open. We have ##z=y## such that there is exactly one point of intersection (because we want it to be tangent). So ##4-x^2 = kx+ k+7##, i.e, ##x^2 +kx + (k+3) = 0##. This yields the criterion ##k^2 - 4(k+3)=0## for ##k##.
K is open? Do you mean variable?
 
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  • #8
WWGD said:
K is open? Do you mean variable?
Yes, of course. That's just the slope of all lines that pass through the point (-1,7).
 
  • #9
DaveE said:
Yes, of course. That's just the slope of all lines that pass through the point (-1,7).
Still, isn't the slope for any such line described by the derivative at the point? Let me reread the op; I may be missing something.
 
  • #10
WWGD said:
Still, isn't the slope for any such line described by the derivative at the point? Let me reread the op; I may be missing something.
One may use derivative, too. Any line passing through ##(a,b)## is of the form ##y-b = k(x-a)##, where ##k## is the slope of the line.
 
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  • #11
nuuskur said:
One may use derivative, too. Any line passing through ##(a,b)## is of the form ##y-b = k(x-a)##, where ##k## is the slope of the line.
...but isn't that the approach that I used? Or how different is your approach?
 
  • #12
Truth be told, I don't really understand what you have done. You write out equalities but don't explain what they are or what you aim for. My crystal ball is not working, either.
 
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  • #13
nuuskur said:
Truth be told, I don't really understand what you have done. You write out equalities but don't explain what they are or what you aim for. My crystal ball is not working, either.
Kindly check my post ##1##. Then let me know if you still don't understand what I did.
 
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  • #14
I don't have this back and forth with my students where they patronisingly tell me to re-read their work and "see if I still don't understand it".

This is not the goal. The goal is for You to understand and convincingly show that you have Understood. You may assume that I have read your attempt. It is not convincing.

##\dfrac{y-7}{x+1}=-2x##
I can infer what this means but it is not the same as you explicitly telling what it means.
and given that, ##y=4-x^2## then;
##4-x^2-7=-2x^2-2x##
Again, I understand that you substitute for ##y## in the first equality, but what is the significance of it?

In short, I'm not doubting your ability to make algebraic manipulations. I'm more interested in why you do them.

This exercise is a trivial one, so you might think it redundant to explain things. But you will carry this attitude to solutions for more complicated problems, which is undesirable.
 
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  • #15
nuuskur said:
I don't have this back and forth with my students where they patronisingly tell me to re-read their work and "see if I still don't understand it".

This is not the goal. The goal is for You to understand and convincingly show that you have Understood. You may assume that I have read your attempt. It is not convincing.
Interesting I may say...I hear you...am not a student either but I always give my students the benefit of doubt...I also can learn from them because I know that I don't know everything.My steps are clear and I fully understand the problem ...thanks for your comments though.
 
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  • #16
nuuskur said:
I don't have this back and forth with my students where they patronisingly tell me to re-read their work and "see if I still don't understand it".

This is not the goal. The goal is for You to understand and convincingly show that you have Understood. You may assume that I have read your attempt. It is not convincing.I can infer what this means but it is not the same as you explicitly telling what it means.

Again, I understand that you substitute for ##y## in the first equality, but what is the significance of it?

In short, I'm not doubting your ability to make algebraic manipulations. I'm more interested in why you do them.

This exercise is a trivial one, so you might think it redundant to explain things. But you will carry this attitude to solutions for more complicated problems, which is undesirable.
The two straight lines and the curve are intersecting at the point ##(x,y)## that's why i made the substitution.
 

Related to Find the coordinates of intersection between tangents and given curve

What is the purpose of finding the coordinates of intersection between tangents and a given curve?

The purpose of finding the coordinates of intersection between tangents and a given curve is to determine the point(s) where the tangent line(s) and the curve intersect. This information is useful in various mathematical and scientific applications, such as optimization problems and curve fitting.

How do you find the coordinates of intersection between tangents and a given curve?

To find the coordinates of intersection, you first need to find the equations of the tangent lines at the point(s) of intersection. This can be done by taking the derivative of the given curve and plugging in the x-coordinate of the point of intersection. Then, set the equations of the tangent lines equal to each other and solve for the x-coordinate. Finally, plug the x-coordinate back into the original curve equation to find the corresponding y-coordinate.

What is the significance of the slope of the tangent line at the point of intersection?

The slope of the tangent line at the point of intersection represents the instantaneous rate of change of the curve at that point. It can also be interpreted as the rate of change of the curve at that point if the curve represents a real-world scenario. Additionally, the slope of the tangent line can be used to determine the direction of the curve at the point of intersection.

Can there be more than one point of intersection between tangents and a given curve?

Yes, there can be more than one point of intersection between tangents and a given curve. This can occur when the curve has multiple peaks or valleys, or when the tangent lines intersect the curve at different angles. In some cases, the curve may also intersect itself, resulting in multiple points of intersection.

What are some real-world applications of finding the coordinates of intersection between tangents and a given curve?

Finding the coordinates of intersection between tangents and a given curve can be applied in various fields such as engineering, physics, and economics. For example, in engineering, this information can be used to optimize the design of a structure or machine. In physics, it can be used to analyze the motion of an object. In economics, it can be used to determine the maximum profit or minimum cost in a production process.

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