The parabola (y+1)^2=a(x2) passes through point (1,2). The equation of its directrix is a) 4x+1=0 b) 4x1=0 c) 4x+9=0 d) 4x9=0
Question
The parabola $(y+1)^2=a(x2)$ passes through point $(1,2)$. The equation of its directrix is
a) $4x+1=0$
b) $4x1=0$
c) $4x+9=0$
d) $4x9=0$
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Answer
To determine the equation of the directrix for the given parabola $(y+1)^2 = a(x2)$, we need to follow these steps:

Substitute the given point $(1, 2)$ into the equation to find the value of $a$.

Rewrite the equation in its standard form to identify the parameters needed to determine the directrix.
Here are the detailed steps:
Step 1: Substitute the Given Point
The equation of the parabola is:
$$ (y+1)^2 = a(x2) $$
Since the parabola passes through the point $(1, 2)$, we substitute $x = 1$ and $y = 2$ into the equation:
$$ (2+1)^2 = a(12) $$
Simplifying that:
$$ (1)^2 = a(1) $$
$$ 1 = a $$
So,
$$ a = 1 $$
Step 2: Rewrite the Equation
Now we substitute $a = 1$ back into the original equation:
$$ (y+1)^2 = 1(x2) $$
Or,
$$ (y+1)^2 = (x2) $$
To bring it into a more familiar form, we reformat it as:
$$ (y+1)^2 = 1(x2) $$
This resembles the standard form of the parabola:
$$ (y  k)^2 = 4p(x  h) $$
Comparing, we can see: $$ h = 2, \quad k = 1, \quad \text{and} \quad 4p = 1 \implies p = \frac{1}{4} $$
Step 3: Equation of the Directrix
The directrix of a parabola of the form $(y  k)^2 = 4p(x  h)$ is given by the line:
$$ x = h  p $$
Substituting $h = 2$ and $p = \frac{1}{4}$:
$$ x = 2  \left(\frac{1}{4}\right) $$
$$ x = 2 + \frac{1}{4} $$
$$ x = \frac{8}{4} + \frac{1}{4} $$
$$ x = \frac{9}{4} $$
Thus, the equation of the directrix is:
$$ x = \frac{9}{4} $$
Rewriting it in the standard form:
$$ 4x  9 = 0 $$
Therefore, the equation of the directrix is:
$$ \boxed{4x  9 = 0} $$
Conclusion
The correct answer is option d) 4x  9 = 0.
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