Respuesta :
Answer:
The comet might intersect the orbiting planet at (0, -4),[tex](\pm\frac{\sqrt{31}}{2},\frac{15}{4})[/tex].
Step-by-step explanation:
Given that, a planet follows an elliptical path described by
[tex]16x^2+4y^2=64[/tex] .........(1)
A comet follows the parabolic path
[tex]y=x^2-4[/tex]
[tex]x^2=y+4[/tex].........(2)
To find the intersecting point, we need to solve the above equations.
Putting [tex]x^2=y+4[/tex] in the equation (1)
[tex]y+4+4y^2=64[/tex]
[tex]\Rightarrow 4y^2+y+4-64=0[/tex]
[tex]\Rightarrow 4y^2+y-60=0[/tex]
[tex]\Rightarrow 4y^2+16y-15y-60=0[/tex]
[tex]\Rightarrow 4y(y+4)-15(y+4)=0[/tex]
[tex]\Rightarrow(y+4)(4y-15)=0[/tex]
[tex]\Rightarrow y=-4, \frac{15}{4}[/tex]
When y = -4 , then [tex]x^2=-4+4[/tex] [tex]\Rightarrow x=0[/tex]
When [tex]y=\frac{15}{4}[/tex], then [tex]x^2=\frac{15}{4}+4[/tex] [tex]\Rightarrow x=\pm \frac{\sqrt{31}}{2}[/tex]
The comet might intersect the orbiting planet at (0, -4),[tex](\pm\frac{\sqrt{31}}{2},\frac{15}{4})[/tex].
