Geometry of Love

Geometry Level 3

A boy named Alex plans to give a rose with a beautiful ring to his sweetheart Rosie.

But Rosie likes a specific kind of rose with a ring such that -

If the rose is considered as a 2-D circle , the ring is a point \(R\) with coordinates \((-3 , -5)\), the power of the point \( R\) (ring) is '101' with respect to the circle(rose). [2- D Geometry]

Alex collects 5 roses with one ring each with coordinates \((-3 , -5)\) and opts to select the appropriate rose, which his sweetheart Rosie would like on Valentine's Day.

The collected roses had the following equations:

Rose R : \(x^2 + y^2 - 2x - 4y + 9 = 0 \)

Rose O : \(x^2 + y^2 - 4x - 6y + 9 = 0 \)

Rose S : \(x^2 + y^2 - 6x - 8y + 9 = 0 \)

Rose I : \(x^2 + y^2 - 8x - 10y + 9 = 0 \)

Rose E :\( x^2 + y^2 - 10x - 12y + 9 = 0 \)

You have to help Alex to select the right rose for his valentine. Which one will you suggest?

Bonus question: Alex and Rosie are the main romantic characters in a film released in 2014. What is the name of this film?

Image Credit: Wikimedia Hamachidori
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