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Question: A comet follows the hyperbolic path described by \(\dfrac{{{x^2}}}{4} - \dfrac{{{y^2}}}{{19}} = 1\),...

A comet follows the hyperbolic path described by x24−y219=1\dfrac{{{x^2}}}{4} - \dfrac{{{y^2}}}{{19}} = 1, where x and y are in millions of miles. If the sun is the focus of the path, how close to the sun is the vertex of the path?

Explanation

Solution

In the above question, we have to find the difference of the distance of the focus from the origin and the vertex from the origin. If we compare the above equation with the equation of hyperbola, we can find the value of a and b and then we can find the value of eccentricity using the relation b2=a2(e2−1){b^2} = {a^2}\left( {{e^2} - 1} \right).

Complete step-by-step answer:
In the above question, it is given that the comet follows a hyperbolic path described as x24−y219=1\dfrac{{{x^2}}}{4} - \dfrac{{{y^2}}}{{19}} = 1 and the sun is at the focus of this hyperbola.
Now, we know that the standard equation of a hyperbola is x2a2−y2b2=1\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1, and on comparing this equation with the above equation, we get
a2=4 , b2=19{a^2} = 4\,,\,{b^2} = 19
Now, put these values in equation b2=a2(e2−1){b^2} = {a^2}\left( {{e^2} - 1} \right).
19=4(e2−1)19 = 4\left( {{e^2} - 1} \right)
⇒e2−1=194\Rightarrow {e^2} - 1 = \dfrac{{19}}{4}
⇒e2=1+194\Rightarrow {e^2} = 1 + \dfrac{{19}}{4}
⇒e2=4+194\Rightarrow {e^2} = \dfrac{{4 + 19}}{4}
⇒e2=234\Rightarrow {e^2} = \dfrac{{23}}{4}
⇒e=234=232\Rightarrow e = \sqrt {\dfrac{{23}}{4}} = \dfrac{{\sqrt {23} }}{2}
In this question, we have to find the difference of the distance of the focus from the origin and the vertex from the origin.
Distance of the focus from the origin =ae = ae
Distance of the vertex from the origin =a = a
Now we have to find the difference,
⇒ae−a\Rightarrow ae - a
⇒a(e−1)\Rightarrow a\left( {e - 1} \right)
Now, put a=2  and  e=232a = 2\,\,and\,\,e = \dfrac{{\sqrt {23} }}{2}
⇒2(232−1)\Rightarrow 2\left( {\dfrac{{\sqrt {23} }}{2} - 1} \right)
⇒23−2 million miles\Rightarrow \sqrt {23} - 2\,million\,miles

Note: Here we have used the center as the origin because this is a standard parabola. If it was not a standard parabola, then we have to find the distance using distance formula with the help of coordinates of focus, vertex and the center.