Question

Write the conditions so that x²+y²−2x−c=0 and x²+y²−2by−c=0 touch externally:

(a) b²−c=0 and a=b
(b) b²−c=a²−c and a=b
(c) a²+b²−c=0 and a=b
(d) a²−b²+c=0 and a=b

Answer 1

The conditions for the two given circles to touch externally are that the sum of the radii equals the distance between the centers, which leads to the conditions (a²+b²−c=0) and (a=b), corresponding to option (c).

Step-by-step explanation:

When two circles defined by the equations x²+y²−2x−c=0 and x²+y²−2by−c=0 touch externally, the mathematical condition represents the fact that the distance between the centers of the circles is equal to the sum of their radii.

Analyzing the given equations, we can see that the center of the first circle is at (1,0) with a radius of √(1²+c), and the center of the second circle is at (0,b) with a radius of √(b²+c). For these two circles to touch externally, the sum of their radii should be equal to the distance between their centers which we calculate using the Pythagorean theorem.

The correct conditions for the circles to touch externally would therefore be (a²+b²−c=0) and the radii being equal (a=b). This corresponds to option (c). The condition a²+b²−c=0 comes from setting the sum of radii equal to the distance between the centers, and a=b comes from the requirement for the radii to be equal.