Focus is $(1,1)$ and equation of the tangent is $x+y=1$.
With this information, we have to find equation of parabola and length of latus rectum.
(Solution is given as $x² -2xy+y² -4x -4y+4=0$)
[Math] Find equation of Parabola when Focus and tangent to the vertex is given.
conic sections
Related Solutions
Let $BE$ be the directrix
As the axis of the parabola is perpendicular to the latus rectum, the equation of the axis(OA) $y=3$
So, $c=a=3$
As the vertex O$(2,3)$ is the mid-point of $A,B, \frac{b+4}2=2\implies b=0$
So, the equation of the directrix(BE) will be $x=0$ as it is perpendicular to the axis.
If P$(h,k)$ be any point on the parabola,
the distance of $AP=\sqrt{(h-4)^2+(k-3)^2}$
the distance of P$(h,k)$ from the directrix $BE$ will be $|h|$
As the eccentricity of a parabola is $1$
So, $$\sqrt{(h-4)^2+(k-3)^2}=|h|$$
Squaring we get $(h-4)^2+(k-3)^2=h^2\implies (k-3)^2=h^2-(h-4)^2=8h-16=8(h-2)$
So, the equation of the parabola : $(y-3)^2=8(x-2)$
The reflection of a parabola's focus in any tangent line gives a point on the directrix. (Why?) Therefore, if you have any two tangents (regardless of the angle they make with one another), then you get two reflected foci, which in turn determine the directrix. With a focus and a directrix, you have a unique parabola. $\square$
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Best Answer
A parabola’s vertex is midway between its focus and directrix. You’re given the tangent line at the vertex, which is parallel to the directrix, so it should be a straightforward matter to find an equation for the directrix: you have its slope/normal and you can find a point on it by reflecting the given focus point $(x_f,y_f)$ in the tangent line. If you put the resulting equation in the form $ax+by+c=0$, then an equation for the parabola is $$(x-x_f)^2+(y-y_f)^2={(ax+by+c)^2\over a^2+b^2}.$$ This equation just expresses the definition of a parabola as the set of points that are equidistant from the focus and directrix. Rearrange this into whatever form is required.
As for the latus rectum, you don’t even need the equation of the parabola to compute its length. You can get the focal distance from the information given, and the latus rectum length is a fixed multiple of this distance.