﻿Hi, I'm Miss Davis. In this lesson, we're going to be looking at the intersections of lines and circles. How many times does a straight line intersect a circle? In this first example, we have got two intersections. We can see that the straight line cuts through the circle at two points. In this next example, there is only one intersection, as the straight line only touches the graph of the circle at one point. In this final example, there are zero intersections, as there is no point on the circumference of the circle where the straight line meets it. The graph of x squared add y squared is equal to nine is shown. We've been asked to find estimates for the solutions of the simultaneous equations x squared add y squared is equal to nine and y is equal to x add 2. We're going to start by plotting the graph of y is equal to x add two. This line will have a gradient of one and a y intercept of two. We're then looking for the point of intersection. We have been asked to estimate the solutions. This means that the coordinates aren't going to be exact. As we can see the top, right coordinate isn't exactly on one, three, we can say that these two points of intersection are -2.9, -0.8 and 0.8, 2.9. Here is a question for you to try. Pause the video to complete your task and resume once you're finished. Here are the answers. When questions require you to read coordinates from a graph, you are given some leeway. If you are 0.1 either side of these points, then that should be fine. However, these values would not satisfy the equation. With this next example, we haven't been given a graph. We're still going to find the two points of intersection of the circle x squared add y squared is equal to 72 and the straight line y is equal to x. We're going to start by substituting y equals x into our equation of the circle. This means that x squared add x squared is equal to 72. We can simplify the left-hand side to say that 2x squared is equal to 72. This means that x squared is equal to 36, and that x is either six or negative six. When x is equal to six, y is also equal to six. When x is equal to negative six, y is also equal to negative six, as we know that x is equal to y or y is equal to x. The points of intersection are six, six, and negative six, negative six. Here is a question for you to try. Pause the video to complete your task and resume once you're finished. Here is the answer. The points of intersection between the circle and the straight line are five, negative five and negative five, five. With this next example, we have been given the two equations of x squared add y squared is equal to 40, and y is equal to 3x add 20. We're going to start by substituting y is equal to 3x add 20 into the equation x squared add y squared is equal to 40. This gives us x squared add 3x add 20 squared is equal to 40. If we expand that second bracket, we get 9x squared add 120x add 400. This simplifies to 10x squared add 120x add 360, as we've made our equation equal to zero by subtracting 40 from both sides. We can divide all of these terms by 10 to give us the simpler equation, x squared add 12x add 36, as all three of the coefficients and the constant are multiples of 10. We can then factorise the left-hand side of this equation to give us the x add six multiplied by x add six is equal to zero. This means that x must be equal to negative six. We can then substitute this into one of the equations that we were given to find that when x is negative six, y is two. This means that the point of intersection is negative six, two. Here is a question for you to try. Pause the video to complete your task and resume once you are finished. Here is the answer. The point of intersection between the circle and the line is 10, negative five. In this next example, we have been asked to show that the line y is equal to 2x subtract 32 does not intersect the graph of x squared at y squared is equal to 100. We're going to start by substituting the first equation of the linear graph into the equation of the circle. This gives us x squared add 2x subtract 32 squared is equal to 100. If we expand that second bracket, it gives us x squared add 4x squared subtract 128x add 1,024 is equal to 100. This simplifies to 5x squared subtract 128x add 924. We're going to use the quadratic formula to solve this quadratic equation. We can see that the value of a is five. The value of b is -128 and c is 924. If we substitute this into the quadratic formula, it gives us this. Negative numbers cannot have a square root. This means that there are no solutions for x which satisfy both equations. This means that there are no points of intersection between the graphs. Here's some questions for you to try. Pause the video to complete your task and resume once you're finished. Here are the answers. The first pair of equations only intersect at one point. The second pair of equations don't intersect at all, and the third set of equations intersect in two places. That's all for this lesson. Thanks for watching.

In this lesson will will investigate the intersection of a line and a circle.

Maths

Y11 Maths (Higher)

Key Stage 4

Year 11

Miss

Intersection of lines and circles

Circle Graphs

Algebra (A)

THEME

UNIT

Draw and recognise circle graphs of the form x² + y² = r²

Decide whether a point lies, on, outside or inside a circle

Find the equation of a tangent to a circle at a given point

Lesson overview:Intersection of lines and circles

Core Content

Circle Graphs: