Question

Assertion: Polar equation of the directrix of the conic \[\dfrac{6}{r} = 1 + 3\cos \theta \] is \[\dfrac{6}{r} = 3\cos \theta \]
Reason: Polar equation of the directrix of the conic \[\dfrac{1}{r} = 1 + e\cos \theta \] is \[\dfrac{1}{r} = e\cos \theta \]
The correct answer is
A.Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B.Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C.Assertion is correct but Reason is incorrect
D.Assertion is incorrect but Reason is correct

Answer 1

Hint: Here we need to choose the correct option. We will first write the general equation of the conic section in polar form and then we will write the general equation of the directrix of the conic section. Then we will compare the given equation of the conic section in polar form with the general equation and also the equation of the directrix of the conic section in polar form with the general equation. From there, we will get our required answer.

Complete step-by-step answer:
We know that the general equation of the conic section in polar form is given by \[\dfrac{1}{r} = 1 + e\cos \theta \] .
Also, general equation of the directrix of the conic section in polar form is given by \[\dfrac{1}{r} = e\cos \theta \], where, \[r\] is distance from centre of the conic and \[e\] is the eccentricity.
The given polar equation of the conic section is \[\dfrac{6}{r} = 1 + 3\cos \theta \].
Now we will write it in the standard form.
On comparing this equation of conic section with the general equation, we get
\[e = 3\] and \[\dfrac{r}{6}\] is the distance from the centre of the conic
Substituting the value of distance from the center of the conic and the eccentricity in \[\dfrac{1}{r} = e\cos \theta \], we get
 \[ \Rightarrow \dfrac{1}{{\dfrac{r}{6}}} = 3\cos \theta \]
On further simplification, we get
\[ \Rightarrow \dfrac{6}{r} = 3\cos \theta \]
Therefore, we can say that the given assertion is correct and also the given reason is the perfect explanation of the given assertion.
Hence, the correct option is option A.

Note: Here, we need to know the general equation of the conic section in polar form and the directrix of the conic section in polar form, so that we can easily compare any given equations with it and we can get the values of the eccentricity and distance from the center of the conic easily. Eccentricity tells the extent of a conic section varies from being circular. We know that the eccentricity of a circle is 1 because it is completely circular.