Intereting Posts

How to do $\frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X}$
How many combinations can be made with these rules? (game of Dobble)
If a set is compact then it is closed
Relative homology
Intersection between two planes and a line?
Homology of a simple chain complex
Meaning of vanishing Lie bracket
Solving an initial value ODE problem using fourier transform
Colimit of $\frac{1}{n} \mathbb{Z}$
Enumerating Sylow $2$-subgroups of Dihedral Group (of order $2^{\alpha}k$ for $k$ odd).
Question on $\Pi_{n=1}^\infty\left(1-\frac{x^a}{\pi^an^a}\right)$ and the Riemann Zeta function
Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$
Prove that series converge
How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?
Calculating CRC by long division: How to decide the top number of long division?

If ($ x_1$,$ y_1$) , ($ x_2$,$ y_2$) & ($ x_3$,$ y_3$) be three points on the parabola $y^2= 4ax$ and the normals at these points meet in a point then prove that $\frac{ x_1 – x_2}{y_3} + \frac{ x_2 – x_3}{y_1} + \frac{ x_3 – x_1}{y_2}$=0.

Normal Equation:

$y=mx-am^3-2am$

- Explicit formula for conformal map from ellipse to unit disc (interior to interior)
- Axis of Symmetry for a General Parabola
- Locus of vertex of moving parabola
- Confusion with the various forms of the equation of second degree
- Equivalence of geometric and algebraic definitions of conic sections
- Intersection of ellipse with circle

Let (x’,y’) be common points

We get:

$y’=mx’-am^3-2am$

Let $m_1, \ m_2$ & $m_3$ be the slopes at ($ x_1$,$ y_1$) , ($ x_2$,$ y_2$) & ($ x_3$,$ y_3$).

We get

$ \ y_1=\ m_1 \ x_1+y’-\ m_1x’$

$ \ y_2=\ m_2 \ x_2+y’-\ m_2x’$

$ \ y_3=\ m_3 \ x_3+y’-\ m_3x’$

To arrive at the desired result i used

$y’=mx’-am^3-2am$

After this step i used $ \ m_1+ \ m_2+ \ m_3=0$ as $\ m^2$ coefficient is ‘0’ but to no avail

- draw $\triangle ABC$ in which $AB=5.5$cm, $\angle C =40^{\circ}$ and $BC-AC=2.5$cm
- How do we define arc length?
- Hexagon packing in a circle
- Why the $O(t^2)$ part in $L(t) = L + t(\csc \alpha_i - \cot \alpha_i +\csc \alpha_{i+1} - \cot \alpha_{i+1}) + O(t^2)$?
- Geometrical Proof of a Rotation
- What is the mathematical significance of Penrose tiles?
- Rotation of a regular tetrahedron
- How to draw an ellipse if a center and 3 arbitrary points on it are given?
- Plot $|z - i| + |z + i| = 16$ on the complex plane
- Approaching the circumference of a circle

Taking a slightly different approach from yours, we have $\nabla(y^2-4ax)=(-4a,2y)^T$, so in homogeneous coordinates the normal through a point $[x:y:1]$ that lies on the parabola is $\mathbf n=[y:2a:-(x+2a)\,y]$. For the normals through three points to have a common intersection, their scalar triple product $\mathbf n_1\times\mathbf n_2\cdot\mathbf n_3$ must vanish, therefore $$\begin{vmatrix} y_1 & 2a & -(x_1+2a)\,y_1 \\ y_2 & 2a & -(x_2+2a)\,y_2 \\ y_3 & 2a & -(x_3+2a)\,y_3 \end{vmatrix} = -2a((x_1-x_2)y_1y_2 + (x_2-x_3)y_2y_3 + (x_3-x_1)y_1y_3)=0$$ which is equivalent to the desired condition if $y_1,y_2,y_3\ne0$.

To put this in terms with which you might be more familiar, if you negate the last column of the above matrix, you have the augmented coefficient matrix of the system of linear equations of the three normals. This system is overdetermined, so for it to have a solution, the rows of the matrix must be linearly dependent, which is equivalent to the determinant of the matrix being zero.

Using $4ax_i = y_i^2$ we can rewrite the statement to be proved as

$\displaystyle\sum_{cyc} y_1y_2(y_1^2-y_2^2) = (y_1-y_2)(y_2-y_3)(y_3-y_1)(y_1+y_2+y_3)$

The parametric form of the Normal at $(at^2,2at)$ is $y+xt = at^3+2at$. If it passes through $(h,k)$, we must have $at^3+(2a-h)t-k=0$

From this we see that if $t_1,t_2,t_3$ are the roots of this equation $\sum t_i=0 \Rightarrow \sum y_i =0$. With this the required statement is readily seen to be true.

- How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$
- Subgroup generated by $2$ and $7$ in $(\mathbb Z,+)$
- Derive an algorithm for computing the number of restricted passwords for the general case?
- Is the restriction map of structure sheaf on an irreducible scheme injective?
- $f(A\cap B)=f(A)\cap f(B)$ $\iff$ $f$ is injective.
- The use of conjugacy class and centralizer?
- Finding the number of consecutive objects
- Isometry in $\mathbb{R}^n$
- Proving the irrationality of $\sqrt{5}$: if $5$ divides $x^2$, then $5$ divides $x$
- First order variation and total variation of a function/stochastic process
- What is the maximum overshoot of interpolating splines in $d$ dimensions?
- Quotient of Gamma functions
- How do I prove the following: $f(S\cup T) = f(S) \cup f(T)$
- Parabola Problem
- Name of this convex polyhedron?