A parabola has a focus of F(2,8.5) and a directrix of y=9.5. The point P(x,y) represents any point on the parabola, while D(x,9.5) represents any point on the directrix.
Maria was asked to use the distance formula to write an equation to represent this parabola.
Here is her work:
Step 1: FP=DP
Step 2: (x−2)2+(y−8.5)2−−−−−−−−−−−−−−−−√=(x−x)2+(y−9.5)2−−−−−−−−−−−−−−−−−√
Step 3: x2−4x+4+y2−17y+72.25=y2−19y+90.25
Step 4: x2−4x−14=−4y
Step 5: −14x2+x+72=y
Identify each incorrect step.
Select all answer choices that both state an incorrect step and explain why it is incorrect.
If there is only one incorrect step, select "only" and the answer choice that states and explains the incorrect step.
Step 5 is incorrect because she divided incorrectly.
Step 4 is incorrect because she added the y-terms incorrectly.
Step 3 is incorrect because a previous step is incorrect.
only
Step 5 is incorrect because a previous step is incorrect.
Step 4 is incorrect because a previous step is incorrect.
Step 3 is incorrect because she expanded the first binomial incorrectly.
Step 2 is incorrect because she used the wrong signs for the coordinates of the focus.
I don't understand this at all... please help!
First, you need to learn how to type math.
x^2 is x squared
1/4 is a fraction; 14 is not
step 1 is ok, since that is the definition of a parabola.
step 2 is ok -- just use the distance formula
step 3 is ok -- both sides squared
step 4 is bad: -19+17 = -2, not -4
step 5 is ok if you use correct text characters
Ty...
I actually got that incorrect.. but it was probably my fault... Thanks anyways
To understand the incorrect steps in Maria's work, let's go through each step and analyze them one by one:
Step 1: FP = DP
This step is correct. It states that the distances from any point on the parabola (P) to the focus (F) and the directrix (D) are equal.
Step 2: (x−2)²+(y−8.5)² = (x−x)²+(y−9.5)²
In this step, Maria made an error by equating the equation of the distance formula for the point P with the equation of the distance formula for the point D. The correct comparison should be between the coordinates of P and D.
Step 3: x²−4x+4+y²−17y+72.25 = y²−19y+90.25
Maria expanded the squared terms incorrectly. When squaring (x - 2) and (y - 8.5), she obtained incorrect coefficients for the x and y terms. The correct expansion would be: (x - 2)² = x² - 4x + 4 and (y - 8.5)² = y² - 17y + 72.25.
Step 4: x²−4x−14 = −4y
This step is correct. She correctly rearranged the equation in the standard form for a parabola, where the equation is in the form y = ax² + bx + c.
Step 5: −14x²+x+72 = y
In this step, Maria made an error when dividing the equation by -4. The correct division would yield -14x²/(-4) = 3.5x², x/(-4) = -0.25x, and 72/(-4) = -18. The correct equation should be -14x² - 0.25x - 18 = y.
Based on this analysis, we can conclude that Step 2, Step 3, and Step 5 are all incorrect.
Therefore, the correct answer is:
Step 2 is incorrect because she equated the wrong equations.
Step 3 is incorrect because she expanded the squared terms incorrectly.
Step 5 is incorrect because she divided incorrectly.