Question

The radius of the director circle of the hyperbola$\dfrac{{{x^2}}}{{a\left( {a + 4b} \right)}} - \dfrac{{{y^2}}}{{b\left( {2a - b} \right)}} = 1;{\text{ }}2a > b > 0$ is:
A.${a^2} + {b^2} + 4ab$
B.$a + b$
C.${a^2} + {b^2} + 2ab$
D.$2{\left( {a + b} \right)^2}$

Answer 1

Hint: It is known that the director of a hyperbola is the locus of the point of intersection of two perpendicular tangents to that given hyperbola, whose general equation is given by, ${x^2} + {y^2} = {a^2} - {b^2}$. Thus, all you need to do to find the radius of the director circle is compare the standard equation of the director circle, with the general equation of the circle.

Complete step-by-step answer:
Let us begin with considering the given equation of the hyperbola,
$\dfrac{{{x^2}}}{{a\left( {a + 4b} \right)}} - \dfrac{{{y^2}}}{{b\left( {2a - b} \right)}} = 1;{\text{ }}2a > b > 0$
The equation of the director circle of any general hyperbola is given by;
${x^2} + {y^2} = {a^2} - {b^2}$
Here, instead of ${a^2}$ we have $a\left( {a + 4b} \right)$ and instead of ${b^2}$ we have $b\left( {2a - b} \right)$.
${x^2} + {y^2} = a\left( {a + 4b} \right) - b\left( {2a - b} \right)$
Simplify the right-hand side of the obtained equation of the direction circle.
$ {x^2} + {y^2} = {a^2} + 4ab - 2ab + {b^2} \\
   = {a^2} + {b^2} + 2ab \\
 $
Now, using the formula, ${\left( {r + s} \right)^2} = {r^2} + {s^2} + 2rs$ to simplify the obtained equation further.
${x^2} + {y^2} = {\left( {a + b} \right)^2}......(1)$
Now, it is known that the standard equation of the circle is given by;
${x^2} + {y^2} = {r^2}......(2)$
Let us compare the equation (1) and (2) to find the radius of the director circle for the given hyperbola.
$
  {r^2} = {\left( {a + b} \right)^2} \\
   \Rightarrow r = a + b \\
$
Hence, the radius of the director circle for the given equation of hyperbola is $a + b$. Thus, you can mark option (B) as your correct option.

Note: In order to solve these types of problems or find any particular equations, you should be thorough with the definitions and the standard formula of the conics involved in the question. Like in this question you need to have a good understanding of the standard forms of equations of hyperbola, director circle and the circle. Also, if at all you do not remember the formula then you can also use the method to find the locus of points that form the required director circle. But here you may go wrong in finding the locus, so the best way is to get familiar with the standard terms and equations of conics.