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Chapter 10: Q. 58 (page 644)

Write an equation for each parabola.

Short Answer

Expert verified

The equation of the parabola is $x^{2} = 4 (y + 1)$ .

Step by step solution

01

Step 1. Given information .

Consider the given graph and the points located on it .

02

Step 2. Formula used .

The formula used to find the equation of parabola is ${(x - h)}^{2} = 4 a (y - k)$ .

where h , k are the points of vertex $(h, k) = (0, - 1)$ and focus $(x, y) = (2, 0)$ .

03

Step 3. Find the equation of parabola .

Substitute the value of $(h, k) = (0, - 1)$ and $(x, y) = (2, 0)$ in the equation of parabola .

localid="1646930530938" ${(x - h)}^{2} = 4 a (y - k) {(2 - 0)}^{2} = 4 a (0 + 1) a = 1$ $y = a {(x - h)}^{2} + k 0 = a {(2 - 0)}^{2} + (- 1) a = \frac{1}{4}$

Further simplify .

Substitute the value of a in the equation of parabola .

localid="1646930607016" ${(x - h)}^{2} = 4 a (y - k) {(x - 0)}^{2} = 4 \times 1 (y + 1) x^{2} = 4 (y + 1)$

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Most popular questions from this chapter

2. To complete the square of $x^{2} - 4 x$ ,add _______.

If $θ$ is acute, the Half-angle Formula for the cosine function is $\cos \frac{θ}{2} =_______$ .

A hyperbola for which $a = b$ is called an equilateral hyperbola. Find the eccentricity $e$ of an equilateral hyperbola.

[Note: The eccentricity of a hyperbola is defined in Problem 81.]

Rutherford’s Experiment In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The figure shows a diagram from Rutherford’s paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola.

(a) Find an equation of the asymptotes under this scenario.

(b) If the vertex of the path of the alpha particles is $10$ cm from the center of the hyperbola, find a model that describes the path of the particle.

Find an equation for each ellipse. Graph the equation by hand.

Center at $(2, - 2)$ : vertex at $(7, - 2)$ : focus at $(4, - 2)$ .

See all solutions

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