Now that we have the value of a = -(1)/(3), we substitute it back into the standard equation x² = 4ay to obtain the specific equation of the parabola.

Find the equation of the parabola which is symmetric about the $y$ -axis, and passes through the point $(2,-3)$ .

Answer: The equation of the parabola is $x^2 = -\frac{4}{3}y$ .

Step-by-step solution

Step 1: Identify the standard equation of the parabola

A parabola that is symmetric about the $y$ -axis has its axis of symmetry along the $y$ -axis. Its standard equation is of the form $x^2 = 4ay$ or $x^2 = -4ay$ . Since the parabola passes through $(2, -3)$ , which is in the fourth quadrant, the parabola must open downwards, meaning the equation will be of the form $x^2 = -4ay$ . However, we can use $x^2 = 4ay$ and the sign of $a$ will naturally adjust.

Step 2: Substitute the given point into the equation

The parabola passes through the point $(2, -3)$ . We substitute $x=2$ and $y=-3$ into the standard equation $x^2 = 4ay$ to find the value of $a$ .

Step 3: Solve for 'a'

After substituting the coordinates, we get $4 = 4a(-3)$ , which simplifies to $4 = -12a$ . Dividing both sides by $-12$ gives us the value of $a$ as $-\frac{1}{3}$ .

Step 4: Write the final equation of the parabola

Now that we have the value of $a = -\frac{1}{3}$ , we substitute it back into the standard equation $x^2 = 4ay$ to obtain the specific equation of the parabola.

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Find the equation of the parabola which is symmetric about the yyy -axis, and passes through the point (2,−3)(2,-3)(2,−3).