I have to solve a problem which says to find the equation of the parabola with focus in A(2;1) and vertex in the origin.
Any suggestions are welcome.
[Math] Find the equation of the parabola with focus (2;1) and vertex in the origin.
conic sections
Related Solutions
Two points on a parabola and its focus just fix a pair of parabolas, and we cannot simply assume that their axis will be parallel/perpendicular to the $y$-axis. Let $A,B$ be our points, $F$ our focus. Let $A',B'$ be the projections of $A,B$ on the directrix: clearly, $A'$ belongs to the circle $\Gamma_A$ centered at $A$ through $F$, $B'$ belongs to the circle $\Gamma_B$ centered at $B$ through $F$. The directrix is a common tangent to $\Gamma_A$, $\Gamma_B$, so it goes through the exterior homothetic centre $H$ of $\Gamma_A,\Gamma_B$. $H$ obviously lies on $AB$, and its position on the $AB$ line just depends on the radii of $\Gamma_A$, $\Gamma_B$, i.e. $AF$ and $BF$.
Summarizing:
- Find $AF,BF$ through the Pythagorean theorem;
- Use the previous informations to locate $H\in AB$;
- One of the tangents to $\Gamma_A$ through $A$ is the directrix of the parabola;
- Given the focus and the directrix, there is nothing else to locate: the vertex is just the midpoint of the segment joining $F$ with its projection on the directrix.
Hint: The axis is the line that is perpendicular to the directrix and passes through the focus. The vertex will be the midpoint of the line segment joining the focus and the intersection of the aforementioned lines.
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Best Answer
The axis of the parabola is the straight line passing through the focus $F=(2,1)$ and the vertex $V=(0,0)$. So the equation of the axis $h$ is $y=\dfrac{1}{2}x$.
The directrix is a straight line $d$ orthogonal to the axis and such that its distance from the vertex $Vd$ is the same as the distance between the vertex and the focus $VF$. So the slope of the directrix is $m=-2$ and it pass thorough $P=(-2,-1)$ so we can find its equation that is $2x+y+5=0$.
Now the parabola is the locus of point $X=(x,y)$ such that the distance from the directrix is the same as the distance from the focus, and writing this we find the equation: $$ \dfrac{|2x+y+5|}{\sqrt{5}}=\sqrt{(x-2)^2+(y-1)^2} $$ Squaring you have the equation of the parabola.