Answer: To find the points where the circle intersects the line x=1, we substitute x=1 in the equation of the circle:
2(1)² + 2y² - 5(1) + 7y - 36 = 0
Simplifying, we get:
2y² + 7y - 31 = 0
We can solve this quadratic equation by using the quadratic formula:
y = (-7 ± √(7² - 4(2)(-31))) / (2(2))
y = (-7 ± √225) / 4
y = (-7 ± 15) / 4
So the two possible values of y are:
y = 2 or y = -8/2 = -4
Therefore, the points where the circle intersects the line x=1 are (1, 2) and (1, -4).
Explanation: