Question

The parabola $(y+1)^2=a(x-2)$ passes through point $(1,-2)$. The equation of its directrix is

a) $4x+1=0$

b) $4x-1=0$

c) $4x+9=0$

d) $4x-9=0$

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Answer

To determine the equation of the directrix for the given parabola $(y+1)^2 = a(x-2)$, we need to follow these steps:

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

  2. 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:

$$ (y+1)^2 = a(x-2) $$

Since the parabola passes through the point $(1, -2)$, we substitute $x = 1$ and $y = -2$ into the equation:

$$ (-2+1)^2 = a(1-2) $$

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:

$$ (y+1)^2 = -1(x-2) $$

Or,

$$ (y+1)^2 = -(x-2) $$

To bring it into a more familiar form, we reformat it as:

$$ (y+1)^2 = -1(x-2) $$

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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