find the pole of x-2y+22 =0 with respect to x2+y2-5x+8y+6=0
Answers
Answer:
First circle - solve by completing the square:
x²+ y² - 4x - 6y - 12 = 0
(x² - 4x) + (y² - 6y) - 12 = 0
(x² - 4x + 4) + (y² - 6y + 9) - 25 = 0
(x-2)² + (y-3)² = 25
So this circle has its center at the point (2,3) and radius 5.
Do the same for the second circle:
x² + y² + 6x + 18y + 26 = 0
(x² + 6x) + (y² + 18y) + 26 = 0
(x² + 6x + 9) + (y² + 18y + 81) - 64 = 0
(x+3)² + (y+9)² = 64
So this circle has its center at the point (-3, -9) and radius 8.
How do we know they touch each other? The x coordinates differ by 5, the y coordinates differ by 12, and the sum of the two radii is 13, and 5/12/13 is a Pythagorean triple. So the radii of the two circles form the hypotenuse of a right triangle, like this:
The point of tangency should be (+1/13, -21/13.) Since the slope of the line that connects the two radii is 12/5, the slope of the tangent line must be -5/12.
Concept
In contrast to basic singularities, like 0 for the logarithm function, and branch points, like 0 for the complex square root function, a pole is a specific form of singularity of a function near which the function behaves reasonably consistently.
Given
Given equation is x- 2y +22 with respect to x2+y2-5x+8y+6 = 0
Find
We need to find the pole
Solution
Let us solve by completing the square first
x²+ y² - 4x - 6y - 12 = 0
(x² - 4x) + (y² - 6y) - 12 = 0
(x² - 4x + 4) + (y² - 6y + 9) - 25 = 0
(x-2)² + (y-3)² = 25
Thus, this circle's radius is 5, and its centre is at the coordinates (2, 3).
Similarly for the second circle
x² + y² + 6x + 18y + 26 = 0
(x² + 6x) + (y² + 18y) + 26 = 0
(x² + 6x + 9) + (y² + 18y + 81) - 64 = 0
(x+3)² + (y+9)² = 64
Thus, this circle has a radius of 8 and a centre at the coordinates (-3, -9)
How can we tell if they made physical contact? A Pythagorean triple (5/12/13) results from the differences in the x and y coordinates, as well as the total of the two radii. In a right triangle, the hypotenuse is created by the radii of the two circles, as seen in the example below:
(+1/13, -21/13) should be the coordinates of the tangent point. The tangent line's slope must be -5/12 since the line that joins the two radii has a slope of 12/5.
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