Answer: (i) To find the range of values of c for which the line intersects the curve at two distinct points, we need to solve the system of equations:
9x² - xy + 4y² = 42 (equation of the curve)
y = 1.5x + c (equation of the line)
Substituting y in terms of x and c, we get:
9x² - x(1.5x + c) + 4(1.5x + c)² = 42
Simplifying and rearranging, we get:
(4.5c - 18)x² + (6c - 12)x + (4c² - 42) = 0
For the line to intersect the curve at two distinct points, the discriminant of this quadratic equation must be positive:
(6c - 12)² - 4(4.5c - 18)(4c² - 42) > 0
Simplifying and solving for c, we get:
-4.5c³ + 27c² - 94.5c + 99 < 0
Using numerical methods (such as a graphing calculator or a computer), we can find that the solution to this inequality is:
1.352 < c < 7.065
Therefore, the range of values of c for which the line intersects the curve at two distinct points is:
1.352 < c < 7.065
(ii) Given that c = 1.5, we need to solve the system of equations:
9x² - xy + 4y² = 42 (equation of the curve)
y = 1.5x + 1.5 (equation of the line)
Substituting y in terms of x and c, we get:
9x² - x(1.5x + 1.5) + 4(1.5x + 1.5)² = 42
Simplifying and solving for x, we get:
x = 2 or x = -1/3
Substituting x in the equation of the line, we get:
y = 4.5 or y = 0.5
Therefore, the coordinates of the two intersection points A and B are:
A: (2, 4.5)
B: (-1/3, 0.5)
(iii) To determine if the point (0.5, 4) lies within the circle, on the circle or outside the circle, we need to find the center and radius of the circle. Since A and B are the endpoints of a diameter, the center of the circle is the midpoint of AB:
x = (2 - 1/3)/2 = 5/6
y = (4.5 + 0.5)/2 = 2.5
Therefore, the center of the circle is (5/6, 2.5). To find the radius of the circle, we can use the distance formula:
r = sqrt((2 - 5/6)² + (4.5 - 2.5)²) = sqrt(5/9 + 4) = sqrt(41/9)
The equation of the circle is then:
(x - 5/6)² + (y - 2.5)² = 41/9
Substituting the coordinates of the point (0.5, 4), we get:
(0.5 - 5/6)² + (4 - 2.5)² = 41/9
Simplifying, we get:
1
Explanation: