Find the equation of the parabola which is symmetric about the y -axis, and passes through the point (2,−3).
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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 x2=4ay or x2=−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 x2=−4ay. However, we can use x2=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 x2=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 −31.
Step 4: Write the final equation of the parabola
Now that we have the value of a=−31, we substitute it back into the standard equation x2=4ay to obtain the specific equation of the parabola.