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1. What is the distance between the center of the circle with equation \(x^2+y^2=-4x+6y-12\) and the point \((1,7)\)?

 

2. The graph of the parabola \(x = 2y^2 - 6y + 3\) has an x-intercept (a, 0) and two y-intercepts (0, b) and (0, c). Find a+b+c.

 

3. The line \(x = 4\) is an axis of symmetry of the graph of  \(y = ax^2 + bx + c\). Find \(\frac{b}{a}\).

 

4. The graph of \(y = ax^2 + bx + c\) is shown below. Find \(a \cdot b \cdot c\). (The distance between the grid lines is one unit.) 

 

5. Geometrically speaking, a parabola is defined as the set of points that are the same distance from a given point and a given line. The point is called the focus of the parabola and the line is called the directrix of the parabola.

Suppose \(\mathcal{P}\) is a parabola with focus (4, 3) and directrix y = 1. The point (8, 6) is on \(\mathcal{P}\) because (8, 6) is 5 units away from both the focus and the directrix.

If we write the equation whose graph is \(\mathcal{P}\) in the form \(y=ax^2 + bx + c\), then what is \(a+b+c\)?

 

Does anyone know how to do any of these? (Possibly all?) Please also include an explanation if you can, thank you!

 Jul 22, 2020
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3. b/a = -4.

 

4. The parabola is y = 1/2*x^2 - 3x + 7/2, so abc = -21/4.

 Jul 23, 2020

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