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Nov 4, 2012 · I'm trying to come up with an equation for determining the intersection points for a straight line through a circle. I've started by substituting the "y" value in the circle equation with the straight line equation, seeing as at the intersection points, the y values of both equations must be identical. This is my work so far:
Solution to Example 1. We first solve the linear equation for y as follows: y = - x - 1/2. We now substitute y in the equation of the circle by - x - 1/2 as follows. (x - 2) 2 + (- x - 1/2 + 3) 2 = 4. We now expand the above equation and group like terms. 2 x 2 - 9 x + 25/4 = 0. Solve the above quadratic equation for x to obtain two solutions.
Higher; Circles and graphs Intersection of a line and circle. The equation of a circle can be found using the centre and radius. The discriminant can determine the nature of intersections between ...
- Intersection Between Circle and Line
- Methods to Find The Point of Intersection of A Line and Circle
- Solved Examples: Intersection Between Circle and Line
- Summary
- FAQs on Circle Line Intersection
A line can intersect a circle in three possible ways, as shown below: 1. We obtain two points of the intersection if a line intersects or cuts through the circle, as shown in the diagram below. We can see that in the above figure, the line meets the circle at two points. This line is called the secant to the circle. 2. If we draw a tangent line to ...
There are two methods to think about this. Method 1: Let us consider the equation of the circle be \({x^2} + {y^2} = {a^2}.\) And that of the line be \(y = mx + c.\) First, if we want to solve the two equations in two unknowns, we need to frame a quadratic equation in \(x.\) Substitute the linear equation in the circle’s equation. Linear equations ...
Q.1. Prove that the line \(y = x + 4\) intersects the circle \({x^2} + {y^2} + 8x + 2y – 84 = 0.\) Ans: We are given a linear equation \(y=x+4.\) The equation of a circle is \({x^2} + {y^2} + 8x + 2y – 84 = 0.\) Substitute \(y = x + 4\) in the equation of the circle \({x^2} + {y^2} + 8x + 2y – 84 = 0.\) \({x^2} + {\left( {x + 4} \right)^2} + 8x + 2...
In this article, we have discussed line and circle and their general forms. Then we saw the three cases of the intersection of a circle and a line. Also, we discussed the two methods of finding the intersection of a circle and a line in detail.
Q.1. What does it mean for a line to intersect a circle at one point? Ans:If a line intersects a circle at only one point, that line will be a tangent to the circle. Q.2. How do you find the intersection of a circle and a line? Ans:We can find the distance of the line from the centre of the circle. If the distance is less than the radius, the line ...
A straight line and circle can have two, one or no points of intersection: If a line and a circle only touch at one point, then the line is a tangent to the circle at that point. To find out how many times a line and circle meet, we can use substitution.
The discriminant Δ = 𝐵 − 4 𝐴 𝐶 of the quadratic 𝐴 𝑥 + 𝐵 𝑋 + 𝐶 = 0 tells us about the intersections of the line and the circle. If Δ> 0, then the line and the circle intersect in two points. If Δ = 0, then the line is tangent to the circle. If Δ <0, then the line and the circle are disjoint.
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Jul 1, 2024 · To find the intersection of a circle and a straight line, solve for x in the linear equation. Substitute the solution for x in the circle equation, and you'll have an easier quadratic equation. These problems can have 0, 1, or 2 solutions, as described in the method above.
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