When two distinct circles lie in the same plane, what is the largest number of points at which they can intersect?
a) 1
b) 2
c) 3
d) 4

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.

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be