The mysterious point

Geometry Level 5

It is found that given any parabola, it is possible to find a point $K$ such that $\frac{l^2}{PK^2}$ + $\frac{l^2}{KQ^2}$ is a constant where $P$ and $Q$ are end points of an arbitrary chord passing through $K$ and $l$ is the length of the semi latus rectum of the parabola. Enter the value of this constant.

This problem is part of my set: Geometry

The answer is 1.

1 solution

Indraneel Mukhopadhyaya
Apr 11, 2016

A short procedure.Let K(a,b).Let inclination of any chord through P be x.Then any point on the chord can be parametrised as x=a+rcosx and y=b+rsinx.Substitute these coordinates in the equation of parabola $y^2$ = $2lx$ so as to get a quadratic in r.The roots of this quadratic equation will be r1=length of PK and r2=length of QK.Now 1/ $r1^2$ +1/ $r2^2$ can be evaluated in terms of x using Veita formulae in the quadratic equation.Now this expression in x is independent of x as it is given to be a constant.Hence equate the value of expression at x=0 and x=pi/2,which will give $a=l$ and b=0.Hence the point K has coordinates ( $l,0)$ and the value of constant comes out to be 1.

The mysterious point

The answer is 1.

1 solution

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