Of the following six points, four are an equal distance from...

Of the following six points, four are an equal distance from the point $P(-1,4)$ and two are not. (a) Identify which four, and (b) Find any two additional points that are the same (nonvertical, nonhorizontal) distance from (-1,4) $Q(-9,10) \quad R(5,12) \quad S(-7,11) \quad T(4,4+5 \sqrt{3})$ $U(-1+4 \sqrt{6}, 6) \quad V(-7,4+\sqrt{51})$

Of the following six points, four are an equal distance from the point $P(-1,4)$ and two are not.
(a) Identify which four, and (b) Find any two additional points that are the same (nonvertical, nonhorizontal) distance from (-1,4) $Q(-9,10) \quad R(5,12) \quad S(-7,11) \quad T(4,4+5 \sqrt{3})$ $U(-1+4 \sqrt{6}, 6) \quad V(-7,4+\sqrt{51})$

Key Concepts

Parameterization of a Circle

Parameterization involves expressing the coordinates of all points on a circle using a parameter, usually an angle. This is achieved by using sine and cosine functions to represent the x and y coordinates, respectively. This concept can be used to generate additional points on a circle that meet extra conditions such as being nonvertical and nonhorizontal.

Circle in the Coordinate Plane

A circle is defined as the set of all points that are at a constant distance (radius) from a fixed point (center). In problems where points are equidistant from a specific point, recognizing that these points lie on a circle is crucial. This concept helps in forming equations and understanding the geometric locus of points that satisfy the distance condition.

Distance Formula

The distance formula is used to calculate the distance between any two points in the coordinate plane. Derived from the Pythagorean theorem, it provides a way to compute this distance as the square root of the sum of the squares of the differences between corresponding coordinates. This concept is essential for determining if multiple points are equidistant from a given point.

Transcript

00:01 This question consists of two parts.

00:02 We will start with part a.

00:04 First we will find distance between point p to q.

00:10 Point p is minus 1 .4 and point q is minus 9 .10.

00:16 We will apply distance formula under root minus 9 minus bracket minus 1 whole square plus 10 minus 4 whole square.

00:27 On solving this we get the value 10.

00:31 Now we will find distance between point p to r.

00:35 Point p is minus 1 .4 and point r is 5 .12 we will apply distance formula here under root 5 minus minus 1 whole square plus 12 minus 4 whole square on solving this we get the value 10 now we will find distance between point p 2 s point p is minus 1 .4 and point s is minus 1 .4 and point s is minus 7.

01:15 We will apply a distance formula here under root minus 7 minus bracket minus 1 whole square plus 11 minus 4 whole square on solving this we get the value under root 85 now we will find distance between point p and t point p is minus 1 .4 and point t is 4 .4 plus 5 root 3 we will apply a distance formula here under root 4 minus bracket minus 1 whole square plus 4 plus 4 plus 5 root 3 minus 4.

02:22 From solving this we get the value 10.

02:25 Now we will find the distance between p and u.

02:30 Point p is minus 1 comma 4 and point u is minus 1 plus 4 root 6 comma.

02:41 6 we will apply distance formula under root minus 1 plus 4 root 6 minus bracket minus 1 whole square plus 6 minus 4 whole square on solving this we get the value 10 now we will apply distance formula between point p and b point p is minus 1 .4 and point b is minus 7 comma 4 and point b is minus 7 comma 4 plus root 51 we will apply distance formula here minus 7 minus bracket minus 1 whole square plus 4 plus root 51 minus 4 whole square...

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