Suppose the circle with equation (x-2)2 +(y+4)2 = 7 is translated by (x, y)➡️(x + 3,y - 1). What is the equation of the image of the
circle? Explain.

The answer to your question is    (x - 5)² + (y + 5)² = 7      

Step-by-step explanation:

Data

Circle equation      (x - 2)² + (y + 4)² = 7

translated (x,y)  (x + 3, y - 1)

Process

1.- Find the center of the circle

Remember that the center is represented as (h, k)

From the standard equation

                                (x - 2)² + (y + 4)² = 7

h = 2

k = -4

Center = (2, -4)

This is the point that is going to change its position

2.- Determine the new coordinates of the center

New center = (2 + 3, -4 - 1)

                   = (5, -5)

3.- Find the new equation of the circle

                       (x - 5)² + (y + 5)² = 7              

The y-intercept of AB is (4/3) and the equation of BC is y = (-1/6) x + (4/3)

If the y-coordinate of point C is 13, its x-coordinate is 4.

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Step-by-step explanation:

AB and BC form a right angle at their point of intersection, B.

So, AB ⊥ BC

A = (14 , -1) , B = (2,1)

Part A: Find the y-intercept of AB.

The general equation of the line y = mx + c

Where: m is the slope and c is constant represents y-intercept.

m = (y₂ - y₁)/(x₂ - x₁)

m =

y = (-1/6) x + c

Substitute with the point (2,1) to find c

1 = (-1/6) * 2 + c

c = 1 + 2 *(1/6) = 4/3

y = (-1/6) x + 4/3

So,  the y-intercept of AB is 4/3

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Part B: Find the equation of BC.

BC ⊥ AB at point B

if m is the slope of AB, the slope of BC = (-1/m)

m = (-1/6)

So, the slope of BC =  (-1/m) = 6

The equation of the line BC ⇒ y = 6x + c

Substitute with the point (2,1) to find c

1 = 6 * 2 + c

c = 1 - 12 = -11

y = 6x - 11

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Part C: Find x-coordinate of C if y-coordinate of point C is 13.

The equation of the BC is y = 6x - 11

Substitute with the y-coordinate of point C = 13

13 = 6x - 11

6x = 13 + 11 = 24

x = 24/6 = 4