Question

The maximum no. of points into which 4 circles and 4 straight lines intersect is (a) 26 (b) 56 (c) 50 (d) 72

Short answer

The maximum number of points into which 4 circles and 4 straight lines intersect is 50. This is calculated by finding the intersection points of straight lines (6), circles (12), and a combination of lines and circles (32), and summing them up (6+12+32=50). Therefore, the correct answer is (c) 50.

Step-by-step solution

Intersection Points in Straight Lines

For the given 4 straight lines, each line can intersect with two others at most. So, there can be a total of 6 pair combinations between those lines leading to their intersection points.

Intersection Points in Circles

For the given 4 circles, each circle can intersect with two others at most since 3 circles cannot intersect at a single point. So, there can be a total of 6 pair combinations between those circles leading to 12 intersection points since one pair of intersection points for each pair of circles.

Intersection Points in Lines and Circles

For the given 4 circles and 4 lines, for each line, it can intersect all the circles. So, each line intersects four circles leading to 2 intersection points for each pair. Thus, total intersection points for every four lines and four circles would be 4 (lines) x 4 (circles) x 2 (intersection points for each pair) = 32.

Calculating the Total Intersection Points

Now, sum up the intersection points from Step 1, Step 2, and Step 3 to get the maximum number of points into which 4 circles and 4 straight lines intersect: Total Intersection Points = Intersection Points in Straight Lines + Intersection Points in Circles + Intersection Points in Lines and Circles Total Intersection Points = 6 + 12 + 32 = 50 So, the maximum number of points into which 4 circles and 4 straight lines intersect is 50. Hence, the correct answer is (c) 50.

Key concepts

Understanding Geometry

Geometry is a branch of mathematics that deals with the study of sizes, shapes, properties, and dimensions of various figures and spaces. At the heart of geometry are the concepts of points, lines, planes, and curves, including circles – which are all considered fundamental constructs. A point has no dimension, it's a mere location. Lines are one-dimensional, stretching infinitely in two directions, and could either be straight or curved.

In our exercise, we've encountered both straight lines and circles, whose intersections are central to solving geometrical problems. When two geometric figures intersect, they share common points, and analyzing intersections helps us understand spatial relationships between figures, which can be essential in fields ranging from architecture to computer graphics.

Circle Intersections

A circle is a perfect round shape, which is defined as the set of all points in a plane that are equidistant from a given center point. The common distance from the center to any point of the circle is called the radius. When we look at circle intersections, we delve into finding the points where two circles meet. Typically, two circles can intersect in two points, one point (if they are tangent to each other), or not at all (if they don't overlap or are concentric).

To calculate the intersection points between circles, as we did in our exercise, involves understanding that each pair could provide two points of intersection. The solution shows that with four circles, there can be six unique pairings, leading to a total of twelve possible intersection points.

Straight Line Intersections

Straight lines are the simplest forms of lines in geometry, infinitely extending in opposite directions without curving. In the context of intersections, when two straight lines meet, they do so at a point, assuming they are not parallel. With our four lines in the exercise, the potential for intersection is worked out by considering the combinations in which the lines can pair up without repeating. Two lines can only intersect once, and so with four lines, we form six unique pairings leading to six discrete intersection points.