Let A = (−7,−4) and B = (7,4), and consider the equation PA•PB = 0. Describe
the configuration of all points P = (x, y) that solve this equation.
Assuming that PA and PB are vectors and PA•PB = 0
is the dot products,
vector PA = (-7-x,-4-y) and PB = (7-x,4-y)
then PA•PB = 0 ---> (-7-x,-4-y)•(7-x,4-y) = 0
-49 + 7x - 7x + x^2 - 16 + 4y - 4y + y^2 = 0
x^2 + y^2 = 65
Which is the equation of a circle, with centre at (0,0) and radius √65
This makes perfect sense, since A and B are the end points of a
diameter. Any point P on a circle joined to the endpoints of a diameter of that circle
creates a right-angled triangle.
Well, it seems like this math problem needs me to put on my thinking cap! So we have two points A = (-7, -4) and B = (7, 4), and we're looking for points P = (x, y) that satisfy the equation PA • PB = 0.
PA • PB = 0 means the dot product of the vectors PA and PB is equal to zero. To understand this, let's break it down a bit. The vector PA is obtained by subtracting the coordinates of point A from point P, and the same goes for the vector PB.
So, PA = (x - (-7), y - (-4)) = (x + 7, y + 4)
And PB = (x - 7, y - 4)
Now, let's calculate the dot product:
(PA • PB) = (x + 7)(x - 7) + (y + 4)(y - 4)
= x^2 - 7x + 7x - 49 + y^2 - 4y + 4y - 16
= x^2 + y^2 - 65
To satisfy the equation PA • PB = 0, we need x^2 + y^2 - 65 to equal zero:
x^2 + y^2 = 65
So, all the points (x, y) that solve this equation are the points lying on a circle centered at the origin (0, 0) with a radius of √65.
In simpler terms, think of it as a circus ring where all the points (x, y) on the ring satisfy the equation!
To describe the configuration of all points P = (x, y) that solve the equation PA•PB = 0, we first need to understand what the equation represents.
The equation PA•PB = 0 represents the dot product of vectors PA and PB, where P is a variable point (x, y) and A and B are fixed points, (-7, -4) and (7, 4) respectively.
Let's break down the steps to solve this equation:
Step 1: Find the vectors PA and PB
To find the vectors PA and PB, we subtract the coordinates of A and B from P.
PA = (x - (-7), y - (-4)) = (x + 7, y + 4)
PB = (x - 7, y - 4)
Step 2: Calculate the dot product
The dot product of two vectors PA and PB is calculated by multiplying their corresponding components and then summing them up.
PA•PB = (x + 7)(x - 7) + (y + 4)(y - 4)
Step 3: Simplify the equation
We can simplify the equation by expanding the terms:
PA•PB = x^2 - 49 + y^2 - 16
Simplifying further, we have:
PA•PB = x^2 + y^2 - 65
Step 4: Set the equation to zero and analyze the configuration
We set the equation equal to zero:
x^2 + y^2 - 65 = 0
This equation represents a circle in the coordinate plane with center at the origin (0, 0) and radius √65.
Therefore, the configuration of all points P = (x, y) that solve the equation PA•PB = 0 is a circle with center (0, 0) and radius √65.
To describe the configuration of all points that satisfy the equation PA · PB = 0, we need to understand what this equation represents geometrically.
First, let's define some terms. In this context, PA represents the vector from point P to point A, and PB represents the vector from point P to point B. The dot product of two vectors, PA · PB, is defined as the product of their magnitudes and the cosine of the angle between them.
So, the equation PA · PB = 0 can be rewritten as |PA| * |PB| * cosθ = 0, where θ is the angle between vectors PA and PB.
Let's consider what happens when a point P satisfies this equation.
If PA · PB = 0, it implies that either |PA| = 0 (i.e., P coincides with point A), or |PB| = 0 (P coincides with point B), or cosθ = 0 (the angle between PA and PB is 90 degrees).
Now, let's examine these cases individually.
1. If P coincides with point A (-7, -4): In this case, PA is the zero vector, so PA · PB = 0 regardless of the value of PB. Therefore, the point (-7, -4) is included in the configuration.
2. If P coincides with point B (7, 4): Similarly, if P is (7, 4), then PB is the zero vector. So again, PA · PB = 0 regardless of the value of PA. Thus, the point (7, 4) is also included in the configuration.
3. If cosθ = 0 (the angle between PA and PB is 90 degrees): In this case, the vectors PA and PB are perpendicular to each other. It means that the slope of the line passing through A and P is the negative reciprocal of the slope of the line passing through B and P.
Let's find the slopes of the two lines:
- The slope of line AB passing through A (-7, -4) and B (7, 4) is (4 - -4) / (7 - -7) = 8/14 = 4/7.
- The negative reciprocal of 4/7 is -7/4.
So, the lines must have slopes of -7/4 to be perpendicular to each other.
This means that all the points P that lie on the line passing through A (-7, -4) with a slope of -7/4, or the line passing through B (7, 4) with a slope of -7/4, satisfy the equation PA · PB = 0.
In conclusion, the configuration of all the points P = (x, y) that solve the equation PA · PB = 0 includes the points (-7, -4), (7, 4), and the lines passing through A and B with a slope of -7/4.