Can someone prove that the minimum distance between two non-intersecting parabolas is along their common normal (without calculus)? (If I understand how to prove it, I'll prove it myself for other curves too.)
2026-04-25 12:32:47.1777120367
Proving that the shortest distance between two parabolas is along their common normal.
343 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRA-PRECALCULUS
- How to show that $k < m_1+2$?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Finding the value of cot 142.5°
- Why is the following $\frac{3^n}{3^{n+1}}$ equal to $\frac{1}{3}$?
- Extracting the S from formula
- Using trigonometric identities to simply the following expression $\tan\frac{\pi}{5} + 2\tan\frac{2\pi}{5}+ 4\cot\frac{4\pi}{5}=\cot\frac{\pi}{5}$
- Solving an equation involving binomial coefficients
- Is division inherently the last operation when using fraction notation or is the order of operation always PEMDAS?
- How is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified like that?
- How to solve algebraic equation
Related Questions in PROOF-WRITING
- how is my proof on equinumerous sets
- Do these special substring sets form a matroid?
- How do I prove this question involving primes?
- Total number of nodes in a full k-ary tree. Explanation
- Prove all limit points of $[a,b]$ are in $[a,b]$
- $\inf A = -\sup (-A)$
- Prove that $\sup(cA)=c\sup(A)$.
- Supremum of Sumset (Proof Writing)
- Fibonacci Numbers Proof by Induction (Looking for Feedback)
- Is my method correct for to prove $a^{\log_b c} = c^{\log_b a}$?
Related Questions in ANALYTIC-GEOMETRY
- Asymptotes of hyperbola
- Position of point with respect to hyperbola
- Length of Shadow from a lamp?
- Show that the asymptotes of an hyperbola are its tangents at infinity points
- Surface by revolution
- All possible values of coordinate k such that triangle ABC is a right triangle?
- Triangle inside triangle
- Is there an equation to describe regular polytopes?
- How do I prove that the gradient between a fixed and any general point on a given line is $m$?
- Three-Dimensional coordinate system
Related Questions in CONIC-SECTIONS
- Show that the asymptotes of an hyperbola are its tangents at infinity points
- Do projective transforms preserve circle centres?
- A Problem Based on Ellipse
- Perfect Pascal Mysticum Points
- I need to rotate this parabola around the y axis, but can't find the correct expression
- Prove that the common chord passes through the origin.
- Rotated ellipse tangent to circle
- tangent to two different branches of the hyperbola
- Probability that a triangle inscribed in an ellipse contains one of its foci
- Locus of mid point of intercepts of tangents to a ellipse
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Assuming the existence of a shortest distance, let the shortest distance join points $A$ and $B$ and construct a circle with diameter $AB$. If the circle cuts either parabola at a point other than $A$ or $B$ there is a line joining the two curves which is shorter than the diameter. Else the circle must be tangent at points $A$ and $B$ and the diameter $AB$ is normal to both curves (because the diameter is perpendicular to the tangent).
There are some implicit assumptions here, but would this do? Note that if $A=B$ then the diameter doesn't have a fixed direction and the situation is different.