Question

which of the following equations define a parabola with a focus at (-10, 10) and a directrix of y = 15? select all that apply. (x + 10)^2 + (y - 10)^2 = (y - 15)^2 y = -\frac{1}{10}(x + 10)^2 + \frac{25}{2} (x - 10)^2 + (y + 10)^2 = (y + 15)^2 (x - 10)^2 + (y + 10)^2 = (y - 15)^2 y = -10(x + 10)^2 - \frac{25}{2} y = -10(x + 10)^2 + \frac{25}{2} y = -\frac{1}{10}(x + 10)^2 - \frac{25}{2}

Explanation:

Step1: Recall parabola definition

A parabola is the set of points \((x,y)\) equidistant from the focus \((h,k)\) and the directrix \(y = d\). The distance formula gives \(\sqrt{(x - h)^2+(y - k)^2}=\vert y - d\vert\). Squaring both sides: \((x - h)^2+(y - k)^2=(y - d)^2\). Here, focus \((- 10,10)\), so \(h=-10,k = 10\), directrix \(y = 15\), so \(d = 15\). Substitute: \((x+10)^2+(y - 10)^2=(y - 15)^2\). This matches the first option.

Step2: Derive vertex form

Expand \((x + 10)^2+(y - 10)^2=(y - 15)^2\):
\((x + 10)^2+y^{2}-20y + 100=y^{2}-30y + 225\)
Simplify: \((x + 10)^2-20y + 100=-30y + 225\)
\((x + 10)^2+10y=125\)
\(10y=-(x + 10)^2+125\)
\(y=-\frac{1}{10}(x + 10)^2+\frac{25}{2}\). This matches the second option.

Step3: Check other options

• Third option: Focus would be \((10,-10)\) (wrong \(h,k\)), directrix \(y=-15\) (wrong \(d\)).
• Fourth option: Incorrect \(h,k\) and \(d\).
• Fifth option: Coefficient wrong (should be \(-\frac{1}{10}\), not \(-10\)), constant term wrong.
• Sixth option: Constant term wrong (should be \(+\frac{25}{2}\), not \(-\frac{25}{2}\)).
• Seventh option: Coefficient and constant term wrong.

Answer:

\((x + 10)^2+(y - 10)^2=(y - 15)^2\), \(y=-\frac{1}{10}(x + 10)^2+\frac{25}{2}\) (i.e., the first and second options:
A. \((x + 10)^2+(y - 10)^2=(y - 15)^2\), B. \(y=-\frac{1}{10}(x + 10)^2+\frac{25}{2}\))