Question

Circles are drawn with diameter being any focal chord of the parabola \[{y^2} - 4x - y - 4 = 0\] will always touch the fixed line then its equation is
A.\[16x - 33 = 0\]
B.\[16x + 33 = 0\]
C.\[8x - 33 = 0\]
D.\[8x + 33 = 0\]

Answer 1

Hint: Here we will first find the basic condition when a circle is drawn with diameter being any focal chord of the parabola. Then we will simplify the given equation of the parabola into the standard form. We will then get the equation of the directrix of the parabola which is the required equation.

Complete step-by-step answer:
Given the equation of the parabola \[{y^2} - 4x - y - 4 = 0\].
As it is given that circles are drawn with diameter being any focal chord of the parabola. So we will use the condition i.e. when a circle is drawn with diameter being any focal chord of the parabola then the circle will touch the directrix of the parabola.
Now we will simplify the given equation of the parabola into the standard equation of the parabola. Therefore, we get
\[ \Rightarrow {y^2} - y = 4x + 4\]
Now we will add \[\dfrac{1}{4}\] on the both side of the equation, we get
\[ \Rightarrow {y^2} - y + \dfrac{1}{4} = 4x + 4 + \dfrac{1}{4}\]
Simplifying the equation, we get
\[ \Rightarrow {\left( {y - \dfrac{1}{2}} \right)^2} = 4x + \dfrac{{17}}{4}\]
Taking 4 common on RHS, we get
\[ \Rightarrow {\left( {y - \dfrac{1}{2}} \right)^2} = 4\left( {x + \dfrac{{17}}{{16}}} \right)\]
Now let the value \[\left( {y - \dfrac{1}{2}} \right)\] be \[Y\] and value \[\left( {x + \dfrac{{17}}{{16}}} \right)\] be \[X\]. Therefore, the equation becomes
\[ \Rightarrow {Y^2} = 4X\]
Now by comparing it to the standard equation of the parabola \[{y^2} = 4ax\], we get the value of \[a\] as \[a = 1\].
Therefore, the equation of the directrix of the parabola is \[X = - 1\]. So by putting the value of \[X\] in this equation we will get the required equation, we get
\[ \Rightarrow x + \dfrac{{17}}{{16}} = - 1\]
\[ \Rightarrow x = - 1 - \dfrac{{17}}{{16}} = - \dfrac{{33}}{{16}}\]
On cross multiplication, we get
\[ \Rightarrow 16x = - 33\]
\[ \Rightarrow 16x + 33 = 0\]
Hence the required equation is \[16x + 33 = 0\].
So, option B is the correct option.

Note: Here we should note that if the equation of the parabola is \[{y^2} = 4ax\], then the parabola is towards the \[x\]-axis and is the equation of the parabola is \[{x^2} = 4ay\], then the parabola is towards the \[y\]-axis. Parabola is a set of points in the Cartesian plane whose distance from a point (focus of the parabola) is equal to the distance from a particular line and this line is generally known as the directrix of the parabola. Directrix of the parabola is always perpendicular to the line of symmetry of the parabola.