Question

For the parabola ${y^2} - 8y - x + 19 = 0$, the focus and the directrix are
$
  a.\left( {\dfrac{{13}}{4},4} \right){\text{ and}}\;x = \dfrac{{11}}{4} \\
  b.\left( {\dfrac{{19}}{7},8} \right){\text{ and}}\;y = 7 \\
  c.\left( {\dfrac{7}{3},3} \right){\text{ and}}\;y = 9 \\
  d.\left( {6,3} \right){\text{ and}}\;x = 7 \\
$

Answer 1

Hint- For the parabola ${Y^2} = 4aX$, coordinates of focus be \[\left( {X = a,\;Y = 0} \right)\] and the equation of directrix be \[X = - a\].

Complete step-by-step solution -
 Given equation of parabola is
${y^2} - 8y - x + 19 = 0$
Now convert this equation into standard form
Now add and subtract by 16 to make a complete square in y.
$
   \Rightarrow {y^2} - 8y + 16 - 16 - x + 19 = 0 \\
   \Rightarrow {\left( {y - 4} \right)^2} = x - 3 \\
   \Rightarrow {\left( {y - 4} \right)^2} = 1\left( {x - 3} \right) \\
$
Now compare above equation with standard form of parabola which is ${Y^2} = 4aX$
$ \Rightarrow Y = \left( {y - 4} \right),{\text{ }}X = \left( {x - 3} \right),{\text{ & }}4a = 1 \Rightarrow a = \dfrac{1}{4}$
The coordinates of the focus of the parabola w.r.t standard parabola is \[\left( {X = a,\;Y = 0} \right)\]
\[
   \Rightarrow \left( {y - 4} \right)\; = 0\;\;\;\;\;\;\;\;\& \;\;\;\;\left( {x - 3} \right)\; = \dfrac{1}{4} \\
   \Rightarrow y = 4\;\;\;\;\;\;\;\;\;\;\;\;\;\& \;\;\;\;\;x = \dfrac{1}{4} + 3 = \dfrac{{13}}{4} \\
\]
So the coordinates of the focus of the given equation is $\left( {\dfrac{{13}}{4},4} \right)$
Now, the directrix of the parabola w.r.t. standard parabola is \[X = - a\]
$
   \Rightarrow x - 3 = - \dfrac{1}{4} \\
   \Rightarrow x = 3 - \dfrac{1}{4} = \dfrac{{11}}{4} \\
$
So, the coordinates of the focus and the equation of directrix is $\left( {\dfrac{{13}}{4},4} \right)$, and $x = \dfrac{{11}}{4}$ respectively.
Hence option (a) is correct.

Note- In such types of questions the key concept we have to remember is that always recall the standard equation of parabola with its standard coordinates of focus and the equation of directrix, then first convert the equation into standard form and compare it with standard equation and calculate the values as above, then calculate the coordinates of focus as above and the equation of directrix which is the required answer.