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Problem 757

Prove that if \(\mathrm{F}(0,1)\) is the focus, and the line \(\mathrm{y}=-1\) is the directrix, then the equation of the parabola is $\mathrm{y}=(1 / 4) \mathrm{x}^{2}$.

Expert verified

To prove that the given parabola has the focus F(0,1) and directrix y = -1, we define a general point P(x,y) on the parabola and find the distances PF and PD between the point and the focus, and the point and the directrix, respectively. Using the definition of a parabola, we equate the two distances, square both sides, and simplify the equation to obtain \(x^2 = 4y\). Dividing by 4, we get the final equation of the parabola as \(y = \frac{1}{4}x^2\).

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Chapter 43

Plot points of the curve corresponding to $\mathrm{y}=(1 / 4)(\mathrm{x}-2)^{2}\( for \)\mathrm{x}=4.3 .2 .1 .0 \mathrm{and}$ sketch the curve.

Chapter 43

Consider the parabola \(y^{2}=4 p x .\) A tangent to the parabola at point \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) is defined as the line that intersects the parabola at point \(\mathrm{P}_{1}\) and nowhere else. (a) Show that the slope of the tangent line is \(2 \mathrm{p} / \mathrm{y}_{1}\) [Hint: Let the slope be \(\mathrm{m}\). Find the equation of the line passing through \(\mathrm{P}_{1}\) with slope \(\mathrm{m}\). What are the points of intersection of the tangent line and the parabola? For what values of \(\mathrm{m}\), would there be only one intersection point?] (b) Find the equation of the tangent line. (c) Prove that the intercepts of the tangent line are $\left(-\mathrm{x}_{1}, 0\right)\( and \)\left[0,(1 / 2) \mathrm{y}_{1}\right]$

Chapter 43

Consider the equation \(x^{2}-4 y^{2}+4 x+8 y+4=0\) Express this equation in standard form, and determine the center, the vertices, the foci, and the eccentricity of this hyperbola. Describe the fundamental rectangle and find the equations of the 2 asymptotes.

Chapter 43

Determine the intercepts, find the asymptotes, and locate the foci of the following hyperbolas: (a) \(x^{2}-\left(y^{2} / 4\right)=1\). (b) \(\left(y^{2} / 16\right)-\left(x^{2} / 4\right)=1\).

Chapter 43

Consider the equation of a parabola $\mathrm{x}^{2}-4 \mathrm{x}-4 \mathrm{y}+8=0$. Find the focus, vertex, axis of symmetry, and the directrix.

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