Understanding Intersection Points: Lines and Circle
The problem asks us to find the absolute maximum number of distinct points where four straight lines and one circle can cross each other. This total count includes points where lines cross lines, and points where lines cross the circle.
Calculating Line-Line Intersections
To get the maximum number of intersections between the four lines themselves, we assume each line crosses every other line at a unique point (no parallel lines, no three lines meeting at one spot).
The formula for the maximum number of intersection points for n distinct lines is given by the combination formula \( \binom{n}{2} \), which calculates how many pairs of lines can be formed.
\( \text{Number of line-line intersections} = \binom{n}{2} = \frac{n(n-1)}{2} \)
With 4 lines (\(n=4\)), we calculate:
\( \binom{4}{2} = \frac{4 \times (4-1)}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 \)
So, the four lines can create a maximum of 6 intersection points among themselves.
Calculating Line-Circle Intersections
Now, let's consider the intersections between the lines and the circle. A single straight line can intersect a circle at most at two distinct points. To achieve the maximum total intersections, we assume each of the four lines intersects the circle at two separate points.
Calculation:
\( \text{Number of line-circle intersections} = (\text{Number of lines}) \times (\text{Max intersections per line with circle}) \)
\( \text{Number of line-circle intersections} = 4 \times 2 = 8 \)
Thus, the four lines create a maximum of 8 intersection points with the circle.
Total Maximum Intersection Points
To find the overall maximum number of intersection points, we sum the maximum points from the line-line intersections and the line-circle intersections.
\( \text{Total Maximum Intersections} = (\text{Max Line-Line Intersections}) + (\text{Max Line-Circle Intersections}) \)
\( \text{Total Maximum Intersections} = 6 + 8 = 14 \)
Therefore, the maximum possible number of intersection points for four straight lines and a circle is 14.
Similar Questions
- Three perfect dice \(D_1, D_2\) and \(D_3\) are rolled. Let \(x, y\) and \(z\) represent the numbers on \(D_1, D_2\) and \(D_3\) respectively. What is the number of possible outcomes such that \(x < y < z\)?
- A man has \(7\) relatives (\(4\) women and \(3\) men). His wife also has \(7\) relatives (\(3\) women and \(4\) men). In how many ways can they invite \(3\) women and \(3\) men so that \(3\) of them are man's relatives and \(3\) of them are his wife's relatives?
- A triangle \(PQR\) is such that \(3\) points lie on the side \(PQ\), \(4\) points on \(QR\) and \(5\) points on \(RP\) respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed ?
- How many four-digit natural numbers are there such that all of the digits are even?
- How many 4-digit numbers are there having all digits as odd?
- In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?
- The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?
- What is the sum of all four digit numbers formed by using all digits \(0, 1, 4, 5\) without repetition of digits?
- In how many ways can a student choose \((n-2)\) courses out of \(n\) courses if 2 courses are compulsory \((n > 4)\)?
If the number of selections of r as well as (n + r) things from 5n different things are equal, then what is the value of r?
Important Questions from Permutation and Combination
The following figure is made with nine squares of length 1 cm each.
A student is standing at point P. The total number of shortest paths to reach point Q from point P, if the student walks along the boundaries of the squares only, is:
Consider the following statements (all the angles are measured in degrees) :
1. If two supplementary angles are $4x + 40$ and $10x$, then $x$ is;
2. The complementary angle of $\frac{4}{5} \times 45$ is :
3. If two complementary angles are in the ratio of $4 : 5$, then the smallest angle is :
Which of the following is correct?
Codes:
1 2 3- There are 40 persons in a palace. If every person shakes hands with every other person, what will be the total number of handshakes?
- How many words can be formed with or without meaning by using all letters of the word DELHI?
A rectangular wall is divided into four squares of equal size where there are two rows each having two squares. The top left square is coloured with green. If, including green, there are three colours available and each square is coloured using any one of these three colours such that no two adjacent squares get painted with the same colour; then how many colour combinations are possible ?
