Intersection Of Line And Circle
Circles and graphs
The equation of a circumvolve can be found using the eye and radius. The discriminant can determine the nature of intersections between two circles or a circle and a line to prove for tangency.
Intersection of a line and circle
At that place are three ways a line and a circumvolve can be associated, ie the line cuts the circle at two distinct points, the line is a tangent to the circle or the line misses the circle.
To piece of work out which case you have, use algebra to work out how many points of intersection there are.
- If the line cuts through the circle, there will exist two points of intersection
- If the line is a tangent to the circle, there volition be one bespeak of intersection
- If the line misses the circle, there will be no point of intersection
Example
The diagram below shows the circle \({10^2} + {y^two} + 18x + 20y + 81 = 0\) and three lines:
- \(y = 10 + 1\) which appears to cut the circumvolve in two points
- \(x = 1\) which appears to be a tangent to the circumvolve
- \(y = - x + 3\) which appears to miss the circle
Here'southward the algebra which should ostend our observations.
The method is substitution.
\[y = x + 1\]
\[{x^2} + {y^2} + 18x + 20y + 81 = 0\]
\[{10^2} + {(x + i)^2} + 18x + 20(10 + 1) + 81 = 0\]
Multiply out the brackets and collect terms.
\[{x^2} + {x^2} + 2x + one + 18x + 20x + 20 + 81 = 0\]
\[2{x^two} + 40x + 102 = 0\]
Factorise the quadratic:
\[2({x^2} + 20x + 51) = 0\]
\[2(x + iii)(x + 17) = 0\]
Therefore \(x = - three, - 17\) and \(y = - 2, - 16\)
Therefore the line \(y = x + 1\) intersects the circle at \(( - 3, - 2)\) and \(( - 17, - sixteen)\) .
Over again using the substitution method.
\[x = ane\]
\[{10^2} + {y^2} + 18x + 20y + 81 = 0\]
\[{1^two} + {y^2} + 18(one) + 20y + 81 = 0\]
Multiply out the brackets and collect terms.
\[1 + {y^two} + 18 + 20y + 81 = 0\]
\[{y^ii} + 20y + 100 = 0\]
Factorise the quadratic:
\[(y + 10)(y + x) = 0\]
\[y = - 10\]
\[x = 1\]
Therefore the line \(x = 1\) intersects the circle at \((1, - ten)\) . So it is a tangent.
Once more using the substitution method.
\[y = - x + three\]
\[{ten^2} + {y^2} + 18x + 20y + 81 = 0\]
\[{x^2} + {( - x + 3)^2} + 18x + 20( - ten + three) + 81 = 0\]
Multiply out the brackets and collect terms.
\[{x^two} + {x^2} - 6x + 9 + 18x - 20x + 60 + 81 = 0\]
\[two{x^two} - 8x + 150 = 0\]
Quadratic does not factorise fully, then find the discriminant:
\[two({x^2} - 4x + 75) = 0\]
\[{b^2} - 4ac = {( - 4)^2} - 4 \times 1 \times 75 = - 284\]
\({b^2} - 4ac\) is negative, therefore there are no real roots.
Therefore the line \(y = - x + 3\) misses the circumvolve.
Intersection Of Line And Circle,
Source: https://www.bbc.co.uk/bitesize/guides/z9pssbk/revision/5
Posted by: dustinhatiltas.blogspot.com
0 Response to "Intersection Of Line And Circle"
Post a Comment