Question

(B) (x - y) +(x + y)? + 6x(x2 - y²)​

Answer 1

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 triplet. 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.