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Chapter 2: Problem 102

Suppose \(b>a>0\). Find a formula in terms of \(y\) for the distance from a typical point \((x, y)\) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) to the point \(\left(0, \sqrt{b^{2}-a^{2}}\right)\).

Short Answer

Expert verified
The short version of the answer is: \[d = \sqrt{a^2 - \frac{a^2 y^2}{b^2} + y^2 - 2y\sqrt{b^2 - a^2} + b^2 - a^2}\]

Step by step solution

01

Recall the distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
02

Plug in the points' coordinates

We have the points \((x, y)\) and \(\left(0, \sqrt{b^2 - a^2}\right)\). So, \((x_1, y_1) = (x, y)\) and \((x_2, y_2) = \left(0, \sqrt{b^2 - a^2}\right)\). We can now substitute the coordinates into the distance formula: \[ d = \sqrt{(-x)^2 + (y - \sqrt{b^2 - a^2})^2} \]
03

Simplify the expression

The distance can be simplified as follows: \begin{align*} d &= \sqrt{x^2 + (y - \sqrt{b^2 - a^2})^2} \\ d &= \sqrt{x^2 + y^2 - 2y\sqrt{b^2 - a^2} + b^2 - a^2} \end{align*}
04

Substitute \(x^2\) with the ellipse equation

From the equation of the ellipse, we have \(x^2 = a^2\left(1 - \frac{y^2}{b^2}\right)\). Let's substitute this into our distance formula: \begin{align*} d &= \sqrt{a^2\left(1 - \frac{y^2}{b^2}\right) + y^2 - 2y\sqrt{b^2 - a^2} + b^2 - a^2} \end{align*}
05

Simplify the expression further

Now we can simplify the expression to get the distance in terms of \(y\): \begin{align*} d &= \sqrt{a^2 - \frac{a^2 y^2}{b^2} + y^2 - 2y\sqrt{b^2 - a^2} + b^2 - a^2} \end{align*} Hence, the distance formula for a typical point \((x,y)\) on the ellipse to the point \(\left(0, \sqrt{b^2 - a^2}\right)\) is: \[ d = \sqrt{a^2 - \frac{a^2 y^2}{b^2} + y^2 - 2y\sqrt{b^2 - a^2} + b^2 - a^2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula
The distance formula is a straightforward and crucial tool in geometry. It helps us find the distance between two points in a coordinate plane. This formula is rooted in the Pythagorean theorem, making it easy to understand. The formula itself is given by:
  • The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
  • In essence, it measures the length of the hypotenuse of a right triangle formed by the x and y differences.
To master this, remember:
  • Square the difference in the x-coordinates.
  • Square the difference in the y-coordinates.
  • Add these squares together.
  • Take the square root of this sum.
This formula is widely used in mathematics, physics, and engineering to determine distances accurately between points, making it a fundamental concept.
Conic Sections
Conic sections are curves obtained by intersecting a plane with a double-napped cone. They are a foundational concept in coordinate geometry, having applications in various fields such as astronomy, physics, and engineering. The primary types of conic sections are:
  • Circle: A special type of ellipse where the two foci coincide.
  • Ellipse: A set of points where the sum of the distances from two distinct foci is constant.
  • Parabola: A set of points equidistant from a point (focus) and a line (directrix).
  • Hyperbola: A set of points where the absolute difference of the distances to two foci is constant.
In this exercise, we are particularly focused on the ellipse, represented by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). It's essential to note:
  • \(a\) and \(b\) are the distances from the center to the vertices along the x and y axes, respectively.
  • The foci are along the major axis, and their separation can be calculated using \(\sqrt{b^2 - a^2}\).
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, combines algebra and geometry to study geometric figures. It utilizes a coordinate plane to define locations and shapes. Here are some core aspects:
  • Every point in the plane is defined by a pair of numbers \((x, y)\).
  • This system simplifies the process of calculating lengths, angles, and areas.
  • It allows expressing geometric problems algebraically.
In coordinate geometry, the ellipse, as used in the problem, can be analyzed by substituting points into its equation to find attributes like distances to foci or axes. The setup consists of:
  • An ordered pair \((x, y)\) representing each point.
  • Equations that describe geometric shapes or paths.
This framework enables us to not only solve geometric problems but also give a geometric interpretation to algebraic equations. Consequently, these concepts bridge the gap between abstract mathematics and concrete geometrical shapes.

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Most popular questions from this chapter

A new snack shop on campus finds that the number of students following it on Twitter at the end of each of its first five weeks in business is 23,89,223 , \(419,\) and \(647 .\) A clever employee discovers that the number of students following the new snack shop on Twitter after \(w\) weeks is \(p(w),\) where \(p\) is defined by $$p(w)=7+3 w+5 w^{2}+9 w^{3}-w^{4}$$ Indeed, with \(p\) defined as above, we have \(p(1)=23,\) \(p(2)=89, p(3)=223, p(4)=419,\) and \(p(5)=647\) Explain why the polynomial \(p\) defined above cannot give accurate predictions for the number of followers on Twitter for weeks far into the future.

Suppose you start driving a car on a chilly fall day. As you drive, the heater in the car makes the temperature inside the car \(F(t)\) degrees Fahrenheit at time \(t\) minutes after you started driving, where $$ F(t)=40+\frac{30 t^{3}}{t^{3}+100} $$ (a) What was the temperature in the car when you started driving? (b) the car ten minutes after you started driving? (c) What will be the approximate temperature in the car after you have been driving for a long time?

Find all real numbers \(x\) such that $$ x^{4}+5 x^{2}-14=0 $$.

Suppose \(M\) and \(N\) are odd integers. Explain why $$ x^{2}+M x+N $$ has no rational zeros.

Explain why the composition of two polynomials is a polynomial.
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