Answer :
Correct answer is (95.45, 3.22) and (74.55, 3.22).
The equation of a circle centered at the origin with radius r is given by the standard form equation x^2 + y^2 = r^2. Since the diameter of the planet's orbit is 140, the radius is 70. Therefore, the equation of the planet's orbit is x^2 + y^2 = 70^2.
The equation of a parabola with directrix x = d and vertex at (h, k) is given by the standard form equation (y - k) = (x - h)^2/(d - h). In this case, the directrix is x = 100 and the vertex is (85, 0). Substituting these values into the standard form equation gives us the equation for the comet's path: (y - 0) = (x - 85)^2/(100 - 85). This simplifies to y = (x - 85)^2/15.
To find the points where the planet's orbit intersects the comet's path, we need to solve the system of equations x^2 + y^2 = 70^2 and y = (x - 85)^2/15 for x and y.
Substituting the second equation into the first equation gives us x^2 + (x - 85)^2/15 = 70^2. Expanding the squared terms and rearranging terms gives us x^2 - 170x + 12100 = 0.
This is a quadratic equation that we can solve using the quadratic formula: x = (170 +/- sqrt(170^2 - 4 * 1 * 12100)) / (2 * 1).
The solutions are x = 95.45 and x = 74.55. These are the x-coordinates of the points where the planet's orbit intersects the comet's path. To find the corresponding y-coordinates we can substitute these values back into either of the original equations. For example, substituting x = 95.45 into y = (x - 85)^2/15 gives us y = (-9.55)^2/15 = 3.22.
Therefore, the points where the planet's orbit intersects the comet's path are (95.45, 3.22) and (74.55, 3.22).
To know more about quadratic equations visit :
https://brainly.com/question/17177510?referrer=searchResults
#SPJ4
