What kind of parabola is that?

Geometry Level 5

We all know that a parabola is defined as the set of points equidistant from a point (the focus) and a line (the directrix). But...

The set of points that are equidistant from the line $y=0$ and the point $(2,3)$ or the point $(2,4)$ can be represented by the equation

$x^4+ax^3+bx^2+cx+dx^2y+fxy+gy^2+hy+j=0$

Find $j$

The answer is 260.

1 solution

Trevor Arashiro
Jun 4, 2016

Working out the two parabolas with foci $(2,3)$ and $(2,4)$ and directrix $y=0$ , the equations are $-x^2+4x+6y-13=0$ and $-x^2+4x+8y-20=0$ respectively.

Simply multiplying the two equations and setting it equal to 0 will yield two parabolas.

$(-x^2+4x+6y-13)(-x^2+4x+8y-20)=0$

The constant term here is equal to j and thus $j=(-13)(-20)=260$

I think it would be clearer to add that the equation encompasses both sets of points.

Julian Poon - 5 years ago

Nice Problem . ! But doesn't deserve level 5 does it?

Prakhar Bindal - 5 years ago

Once you see the trick the problem becomes very easy. However, until then, the just seeing the quartic in X and y throws many, many people off. Also judging from the 6% of solvers, I'd say this is probably a lvl 5 problem.

Trevor Arashiro - 5 years ago

Actually this type of question came in My JEE Preparation test so thats why seemed easy to me because i had already solved it during my test

Prakhar Bindal - 5 years ago

What kind of parabola is that?

The answer is 260.

1 solution

0 pending reports