How do you find the points on the parabola 2x=y^2 that are closest to the point (3,0)?

hotonglamoz 2022-09-20 Answered
How do you find the points on the parabola 2 x = y 2 that are closest to the point (3,0)?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

seguidora1e
Answered 2022-09-21 Author has 8 answers
Points are (2.2) and (2, -2)
Let there be any point (x,y) on this parabola. The distance 's' of this point from point (3,0) is given by s 2 = ( x - 3 ) 2 + y 2 . Differentiate both sides w.r.t x
2 s d s d x = 2 ( x - 3 ) + 2 y d y d x . For minimum distance d s d x =0, hence (x-3)+ y d y d x = 0 ,
Or y d y d x =3-x
Differentiating the equation y 2 =2x with respect to x, it would be y d y d x = 1
It is thus 3-x=1, x=2, and then y=2, -2. The nearest points are (2,2) and (2,-2)

We have step-by-step solutions for your answer!

koraby2bc
Answered 2022-09-22 Author has 2 answers
An alternative starts the same as bp's solution:
Let (x,y) be any point on this parabola. The distance 's' of this point from point (3,0) is given by s 2 = ( x - 3 ) 2 + y 2
Note that, since (x,y) is on the graph, we must have y 2 = 2 x , so
s 2 = ( x - 3 ) 2 + 2 x
Our job now is to minimize the function:
f ( x ) = ( x - 3 ) 2 + 2 x = x 2 - 4 x + 9
(It should be clear that we can minimize the distance by minimizing the square of the distance.)
To minimize, differentiate, find and test critical numbers.
f ( x ) = 2 x - 4 , so the critical number is 2
f ( 2 ) = 2 is positive, so f(2) is a minimum.
The question asks for points on the graph, so we finish by finding points on the graph at which x=2
2 ( 2 ) = y 2 has solutions y = ± 2
The points are: (2,2) and (2,−2)

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-04
What is the Proximal Operator ( Prox) of the Pseudo L 0 Norm?
Namely:
Prox λ 0 ( y ) = arg min x 1 2 x y 2 2 + λ x 0
Where x 0 = n n z ( x ), namely teh number of non zeros elements in the vector x.
asked 2022-06-16
Let x 1 , . . . , x 25 > 0 be such that i = 1 25 x i = 4350 and i = 1 25 x i 2 = 757770.25.
From the first equality alone, we know that at least one of the x i 's must be less than or equal to 4350 25 = 174. From the second equality alone, we know that at least one of the x i 's must be less than or equal to 757770.25 25 = 174.1, which is less useful than the first bound. My question is whether we can get a better bound, i.e. to find the least upper bound of min { x 1 , . . . , x 25 } , when we use both equalities together. I appreciate any comments or hints.
asked 2022-06-20
I have a doubt regarding a constrained optimisation problem.

Suppose my original constrained minimisation problem is
min x f ( g ( x ) , x )  s.t.  g ( x ) = 3
I would like to know if this equivalent to solving the unconstrained minimisation problem
min x f ( 3 , x )
If not, when are these two problems equivalent?
asked 2022-08-11
How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of 8 , 788 ( c m ) 3 ?
asked 2022-08-16
What is a tight lower bound to i = 1 n 1 a + x i under the restrictions i = 1 n x i = 0 and i = 1 n x i 2 = a 2 ?
Conjecture: due to the steeper rise of 1 a + x for negative x, one may keep those values as small as possible. So take n 1 values x i = q and x n = ( n 1 ) q to compensate for the first condition. The second one then gives a 2 = i = 1 n x i 2 = q 2 ( ( n 1 ) 2 + n 1 ) = q 2 n ( n 1 ). Hence,
i = 1 n 1 a + x i n 1 a ( 1 1 / n ( n 1 ) ) + 1 a ( 1 + ( n 1 ) / n ( n 1 ) )
should be the tight lower bound.
asked 2022-09-23
1- What is Optimization? How many methods are there to calculate it?
2- What do we mean by an objective function? What do we mean by constraints?
3- Give three practical examples (physical or engineering) of a target function with a constraint
asked 2022-05-09
Could you help me solve this problem please ?

1. Maximize x t y with constraint x t Q x 1 (where Q is definite positive)
What I tried : I tried using KKT but I don't know why I get y t Q 1 y as the maximum instead of y t Q 1 y (which I believe is the maximum). Also, since x t y is linear (convex and concave), I don't know how to conclude...
2. Conclude that ( x t y ) 2 ( x t Q x ) ( y t Q 1 y ) x , y (generalized CS)

New questions