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Since we are given that the line 𝑦 + 2 𝑥 = 5 is tangent to the circle (𝑥 − 𝑝) + (𝑦 − 2) = 5 , we know that there exists a point at which the line and the circle meet; in particular, at this point, the value of 𝑦 in the equation of the line will be equal to the value of 𝑦 in the equation of the circle.
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understand that a line can intersect a circle at two points,...
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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:
To find out how many times a line and circle meet, we can use substitution. Problem 1 : Find the points where the line with equation y = 3x intersects the circle with equation x 2 + y 2 = 20 .
- Intersection Between Circle and Line
- Methods to Find The Point of Intersection of A Line and Circle
- Solved Examples: Intersection Between Circle and Line
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- 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 ...
If the line cuts through the circle, there will be two points of intersection; If the line is a tangent to the circle, there will be one point of intersection; If the line misses the circle, there ...
Oct 3, 2023 · ∴ The intersection points of the circle and the line are (4 ± √ (76))/10 and (-1 ± √ (76))/5. Conclusion. We have explored the concept of geometric intersection, which is the point, or set of points where geometric shapes or lines meet or cross. We covered specific cases in two and three-dimensional spaces.
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4 days ago · An (infinite) line determined by two points (x_1,y_1) and (x_2,y_2) may intersect a circle of radius r and center (0, 0) in two imaginary points (left figure), a degenerate single point (corresponding to the line being tangent to the circle; middle figure), or two real points (right figure). In geometry, a line meeting a circle in exactly one point is known as a tangent line, while a line ...
