600 followers problem!

Geometry Level 5

If a circle passes through the points of intersection of the coordinate axes with the lines $\lambda x-y+1=0$ and $x-2y+3=0$ , then let the possible values of $\lambda$ be $\lambda_1,\lambda_2,\cdots$ . Calculate $\left\lceil \displaystyle\prod_{i=1}^n \lambda_i\right\rceil+\left\lfloor \displaystyle\sum_{i=1}^n \lambda_i\right\rfloor$

The answer is 2.

1 solution

Pranjal Jain
Apr 29, 2015

The points of intersection of coordinate axes with given lines are $A(-3,0),B(0,\frac{3}{2}),C(0,1),D(\frac{-1}{\lambda},0)$

These $4$ points must be concyclic.

$\text{Case 1:}$ Two of the points are same as $3$ points are always concyclic.

$\text{Case 1.1:}$ $A=D\Rightarrow \frac{-1}{\lambda}=-3\Rightarrow \lambda=\frac{1}{3}$

$\text{Case 1.2:}$ No $D$ exists. That is, first line is parallel to x-axis. $\lambda=0$

$\text{Case 2:}$ $OA\times OD=OB\times OC\Rightarrow \lambda=2$

$PS:$ You may use any other condition as well in Case 2.

$\lceil 0\times 2\times \frac{1}{3}\rceil+\lfloor 0+2+\frac{1}{3}\rfloor=1+2=\boxed{2}$

The problem explicitly states that a circle passes through the points of intersection of each of the given pair of lines. In case of the absence of such a point, shouldn't the case be rejected ?

Aditya Dhawan - 4 years, 7 months ago

if we find the equation of the circle and satisfy the point, we get a quadratic in lambda. It gives answer 3. Are you sure, 2 is the correct answer??

Bharat Gangwani - 1 year, 4 months ago

600 followers problem!

The answer is 2.

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