6
votes
Accepted
Maximum Length of Tangent to Circle
Let $A=(2,3)$, $B=(3,4)$, and $C=(10,11)$. Now, consider two arbitrary circles $\omega_1$ and $\omega_2$ passing through $B$ and $C$, with their tangent points from $A$ being $P$ and $Q$. Since $A$, $...
- 1,321
6
votes
Accepted
The product of the alternating lengths of a hexagon whose vertices belong to an ellipse
First of all, it is enough to prove the property in a circle. That's because we can always apply an affine transformation to squish the ellipse into a circle:
The affine transformation will not ...
- 151k
5
votes
Accepted
Prove or disprove that the envelope of some chords of a conic section is another conic section
Let $f(x,y) = \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1 = 0$ be the equation of the original ellipse. Let $P=(u,v)$ be a point in the ellipse other than $A = (x_0,y_0)$ and let $\ell$ be a line through $P$...
- 3,538
5
votes
Accepted
Approximation of $\pi$ using an ellipse and a triangle
It helps to analyze the approximation quantitatively using orders of infinitesimals. For example, if we approximate the sector of area $\fracθ2$ using only the triangle of area $\frac{\sin θ \cos θ}{2}...
- 9,587
4
votes
Maximum Length of Tangent to Circle
Let $A(3,4)$, $B(10,11)$ and $C(2,3)$.
The fact that $A,B,C$ are aligned is essential.
I have already given an answer to a similar question here.
Here is a qualitative presentation of this result.
...
- 85.7k
4
votes
A Question about the Product of the Slopes of Two Line Segments within an Ellipse
This proof does not use coordinates, it is a synthetic one.
The idea is to reduce the problem to a simpler one about a circle. If we dilate our ellipse vertically with ratio $a/b$, it becomes a circle ...
- 17.4k
4
votes
A Question about the Product of the Slopes of Two Line Segments within an Ellipse
Geometric (but slightly algebra-heavy) Solution
(See the Edit History for a more complicated take.)
Changing some of OP's notation, consider a conic with focus $F$, focal chord $PQ$, and vertex $V$. ...
- 80.3k
4
votes
Accepted
Condition $A^2+B^2\neq 0$ for conic sections?
That condition ensures you don't have a line. Assuming $A$ and $B$ are real, the only way for $A^2 +B^2 = 0$ is if both $A$ and $B$ are $0$ which, according to your equation defining conic sections, ...
- 30.1k
4
votes
Accepted
Relating the eccentricities of the Gergonne-Steiner and Nine-point conics of a quadrilateral
$\newcommand{\sgn}{\operatorname{sgn}}$
Coordinatizing the vertices of $\square ABCD$ via
$$\begin{align}
A=\phantom{-}a(\cos\theta,\sin\theta) &\qquad B=\phantom{-}b(\cos(-\theta),\sin(-\theta)) \...
- 80.3k
3
votes
Accepted
Can this locus be an ellipse?
While dealing with such questions where we have to establish relations between slopes of lines based on angles between them, it must be kept in mind that while using the formula for the acute angle ...
- 4,338
3
votes
A Question about the Product of the Slopes of Two Line Segments within an Ellipse
This is essentially a "Second Addendum" to my previous answer, provided separately to avoid sprawl. Moreover, it offers a generalization to the problem that seems to warrant separate ...
- 80.3k
2
votes
Relating the eccentricities of the Gergonne-Steiner and Nine-point conics of a quadrilateral
Although Blue already gave a very nice proof, I would like to share my approach with a different coordinatization. After deriving the equations of the two conics, I follow the final steps of Blue's ...
- 886
2
votes
Focus of parabola with two tangents
Let the focus be $S(h,k)$ with $A(1,1)$ and $B(1,0)$ the given points of tangency
The image of focus in any tangent lies on the directrix and is in fact the foot of the perpendicular from the point ...
- 3,816
2
votes
Maximum Length of Tangent to Circle
The center of the circle lies on the perpendicular bisector of the segment joining $(3,4)$ and $(10,11)$. Therefore, the center is $O(a,-a+14)$ for some $a\in\Bbb R$ and the equation of the circle is
$...
- 14k
2
votes
Maximum Length of Tangent to Circle
Hint:
WLOG, the equation of the circle $$x^2+y^2+2gx+2fy+c=0$$
where the center $O(-f,-g),$ radius $=\sqrt{f^2+g^2-c}$
As the circle passes through $(3,4);(10,11)$ we can replace the values of $(x,y)$ ...
- 277k
2
votes
The equation of the largest circle with center (1,0) which is inscribed in the ellipse $x^2 + 4y^2 = 16$.
The largest circle will touch the ellipse, hence there will be only one solution for
\begin{align}
(x+1)^2+y^2&=r^2\\
x^2+4y^2&=16
\end{align}
in the upper right quadrant ($x,y>0$).
...
- 5,059
1
vote
Accepted
Convex hexagon $|AB|=|AF|=|AD|=|BE|=|CF|=a$, Prove $a$ reaches maximum when six points are on an ellipse.
Let us see first how to realize a constellation of points as the given one.
We denote by $x$ the angles formed by $AC$ with each of $AB,AD$,
and by $X$ the length of the segment $AC$.
We denote by $y$...
- 36.3k
1
vote
The equation of the largest circle with center (1,0) which is inscribed in the ellipse $x^2 + 4y^2 = 16$.
Equation of circle with centre $(1,0)$ can be assumed as $(x-1)^2+y^2=r^2$
Solving with $x^2+4y^2=16$ we get a quadratic in $x$ :
$(x-1)^2+\frac{16-x^2}{4}=r^2$
i.e. $\frac{3}{4}x^2-2x+5-r^2=0$.
Since ...
- 9,920
1
vote
A rectangular hyperbola passes through the points A(1, 1), B(1, 5) and C(3, 1). The equation of normal to the hyperbola at A(1, 1) is A:
First of all, it is obvious that the hyperbola $ABC$ doesn't have its axes or its asymptotes parallel to the coordinate axes (see figure), so that a direct computation of its equation is quite ...
1
vote
Intuitive explanation of Pascal's Theorem
To provide such an intuitive explanation, I will prove Pascal's theorem and Brianchon's theorem for conics using a central projection (given that the theorems for circles are already proved)
Let $\...
- 886
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