Question

When two distinct circles lie in the same plane, what is the LARGEST number of points at which they can intersect?

Answer 1

The answer should be letter B) 2

Answer 2

Answer: Option B

Step-by-step explanation:

A circle has the same curve always (we do not have any point of inflection)

so you can think this as a quadratic equation, if one of the arms touchs the x-a-axis, then the other arm also must do it.

The case is the same here, we can have a maximum of 2 intersections.

two circles can be written as:

x^2 + y^2 = R^2

centered around the (0,0) of radious R

(x - a)^2 + (y - b)^2 = P^2

centered in the point (a,b) and the radious is P

the fact that we have an intersection means that in both circles we have the pair (x,y)

for this we can do:

x^2 + a^2 - 2ax + y^2 + b^2 - 2by = P^2

now we assume that x and y are solutions of the first circle, so we replace x^2 + y^2 with R^2

R^2 + a^2 + b^2  -2(ax + by) = P^2

2(ax + by) = -P^2 + R^2 + a^2 + b^2

now we replace one of the variables by using:

x^2 + y^2 = R^2

x^2 = R^ - y^2

x = √(R^2 - y^2)

(a√(R^2 - y^2) + by) = (-P^2 + R^2 + a^2 + b^2)/2

this can not be simplified, but is easy to see that the maximum posible degree of the polynomial is 2, so the maximum number of roots of the polinomial can be 2.