The point at which the line 2 x+√6y=2 touches the curve x^2-2y^2=4 is
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How do I determine the equation of the tangent to the curve y=2x−x2y=2x−x2 that passes through point (2,9)(2,9)?
The question is about the methodology ("How do I..." rather than "What is..."), so the answer does not include the final result to be found.
We know that the tangent is, by definition, a straight line, and as such its general expression is y = ax + b.
We also know that, by definition, both the tangent and the curve pass through one single same point. On this point, we have ax + b = (y =) 2x - x^2.
In other words, x^2 + (a - 2)x + b = 0 [0] has a unique solution, which is x = (2 - a)/2. (Look up Discriminant on Wikipedia if necessary).
Note: this last result can also be obtained by using the curves derivative, whose value at the point of contact is a. in other words, 2 - 2x = a, which in turn gives x = (2 - a)/2.
Either way, this means that (re-injecting that solution for x into [0]) we have:
((2 - a)^2)/4 + (a - 2)*(2 - a)/2 + b = 0. [1]
Additionally, since the tangent goes through point (2,9), then we also know that:
9 = 2a + b. [2]
Equations [1] and [2] create a 2-equation system for two unknowns (a and b), which you can solve:
- Replace b in [1] by a function of a obtained from [2];
- Solve [1] for a;
- Use the solution for a in [2] and solve for b.
This will give you respectively the solutions for the slope a and the intercept b of the tangent (spoiler alert: they are small integers)
I hope this will help you
if not then comment me
How do I determine the equation of the tangent to the curve y=2x−x2y=2x−x2 that passes through point (2,9)(2,9)?
The question is about the methodology ("How do I..." rather than "What is..."), so the answer does not include the final result to be found.
We know that the tangent is, by definition, a straight line, and as such its general expression is y = ax + b.
We also know that, by definition, both the tangent and the curve pass through one single same point. On this point, we have ax + b = (y =) 2x - x^2.
In other words, x^2 + (a - 2)x + b = 0 [0] has a unique solution, which is x = (2 - a)/2. (Look up Discriminant on Wikipedia if necessary).
Note: this last result can also be obtained by using the curves derivative, whose value at the point of contact is a. in other words, 2 - 2x = a, which in turn gives x = (2 - a)/2.
Either way, this means that (re-injecting that solution for x into [0]) we have:
((2 - a)^2)/4 + (a - 2)*(2 - a)/2 + b = 0. [1]
Additionally, since the tangent goes through point (2,9), then we also know that:
9 = 2a + b. [2]
Equations [1] and [2] create a 2-equation system for two unknowns (a and b), which you can solve:
- Replace b in [1] by a function of a obtained from [2];
- Solve [1] for a;
- Use the solution for a in [2] and solve for b.
This will give you respectively the solutions for the slope a and the intercept b of the tangent (spoiler alert: they are small integers)
I hope this will help you
if not then comment me
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