(a) Explanation Given Information: The circle is given by the equation x 2 + y 2 = 9 and a point ( 4 , 5 ) . Formula used: To find an equation of a line with the help of given point and center of the given equation of circle to evaluate minimum and maximum distance from an arbitrary point to the circle. A point on circle is closed to a given point when x -coordinate is positive. Standard form of equation of circle is ( x − h ) 2 + ( y − k ) 2 = r 2 where center is ( h , k ) and r is the radius. Equation of a line is given by y − y 1 = m ( x − x 1 ) where m = y 2 − y 1 x 2 − x 1 , ( x 1 , y 1 ) and ( x 2 , y 2 ) are arbitrary points. According to the substitution method: (1) Solve one of the equations for x or y . (2) Substitute the calculated values of x or y in terms of the other variable, in the other equation. (3) Solve the equation. (4) Substitute the calculated value in the first equation to solve for the other variable. (5) Check by substituting the values of x and y in both the equations. Calculation: Consider the equation of circle, x 2 + y 2 = 9 Rewrite above equation as, ( x − 0 ) 2 + ( y − 0 ) 2 = 3 2 Compare above equation to standard form of equation of circle ( x − h ) 2 + ( y − k ) 2 = r 2 where center is ( h , k ) and r is the radius. Thus, the center of the circle x 2 + y 2 = 9 is ( 0 , 0 ) . The given point is ( 4 , 5 ) . Find an equation of line passing through ( 0 , 0 ) and ( 4 , 5 ) . Here, ( 0 , 0 ) is ( x 1 , y 1 ) and ( 4 , 5 ) is ( x 2 , y 2 ) . Equation of a line is given by y − y 1 = m ( x − x 1 ) where m = y 2 − y 1 x 2 − x 1 . Substitute ( 0 , 0 ) for ( x 1 , y 1 ) and ( 4 , 5 ) for ( x 2 , y 2 ) . m = y 2 − y 1 x 2 − x 1 = 5 − 0 4 − 0 = 5 4 Therefore, equation of line is, substitute 5 4 for m and ( 0 , 0 ) for ( x 1 , y 1 ) (b) To determine To calculate: The coordinates of a point on the circle given by x 2 + y 2 = 9 that is farthest to the point ( 4 , 5 ) .

2nd Edition

ISBN: 9780077836344

Author: Julie Miller, Donna Gerken

Publisher: McGraw-Hill Education

See similar textbooks

Concept explainers

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These…

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are…

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) ge…

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.

Videos

Question

Chapter 5.4, Problem 74PE

(a)

To determine

To calculate: The coordinates of a point on the circle given by x2+y2=9 that is closest to the point (4,5).

(b)

To determine

To calculate: The coordinates of a point on the circle given by x2+y2=9 that is farthest to the point (4,5).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.4, Problem 73PE

Chapter 5.4, Problem 75PE

Chapter 5 Solutions

College Algebra (Collegiate Math)

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The coordinates of a point on the circle given by x 2 + y 2 = 9 that is closest to the point ( 4 , 5 ) .

Solution Summary: The author explains how to calculate the coordinates of a point on the circle given by the equation x2+y2.