A line passes through #(9 ,5 )# and #(2 ,3 )#. A second line passes through #(2 ,8 )#. What is one other point that the second line may pass through if it is parallel to the first line?

To determine if circle B overlaps circle A after being translated by <2, -1>, we need to find the distance between the centers of the two circles after the translation and compare it to the sum of their radii. 1. Translate the center of circle B by <2, -1>: New center for circle B = (1 + 2, 7 - 1) = (3, 6) 2. Calculate the distance \( d \) between the centers of circle A and the translated circle B: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] \[ d = \sqrt{(3 - 3)^2 + (6 - 3)^2} \] \[ d = \sqrt{0 + 9} \] \[ d = \sqrt{9} \] \[ d = 3 \] 3. Sum of the radii of circle A and circle B: \( r_1 \) = 3 (radius of circle A) \( r_2 \) = 5 (radius of circle B) \[ r_1 + r_2 = 3 + 5 = 8 \] Since the distance \( d \) between the centers of the two circles after translation (3) is less than the sum of their radii (8), circle B overlaps circle A. To find the minimum distance between points on both circles: 1. Subtract the radii from the distance \( d \): \[ \text{Minimum distance} = d - (r_1 + r_2) \] \[ \text{Minimum distance} = 3 - 8 \] \[ \text{Minimum distance} = -5 \] The minimum distance between points on both circles is 5 units. --- For the second question: Given line passes through the point (2,8) and is parallel to the line passing through (9,5) and (2,3). The slope of the first line \( m \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \[ m = \frac{3 - 5}{2 - 9} \] \[ m = \frac{-2}{-7} \] \[ m = \frac{2}{7} \] Using the point-slope form \( y - y_1 = m(x - x_1) \), the equation of the second line is: \[ y - 8 = \frac{2}{7}(x - 2) \] To find another point on this line, you can choose any x-value and plug it into the equation to find the corresponding y-value. For example, let's use \( x = 5 \): \[ y - 8 = \frac{2}{7}(5 - 2) \] \[ y - 8 = \frac{2}{7}(3) \] \[ y - 8 = \frac{6}{7} \] \[ y = \frac{6}{7} + 8 \] \[ y = \frac{6}{7} + \frac{56}{7} \] \[ y = \frac{62}{7} \] So, another point that the second line may pass through is (5, \( \frac{62}{7} \)).