if the equation x2 + y2 - 8x +12y = -2 represents a circle

Question

1 Center and radius
2 Equation of a tangent to the circle through (5, 1)
2 Equation of a chord that coincides with the diameter and includes (5, 1)
2 End point coordinates of the preceding

Answer 1

The equation of a circle is

 

(x - h)² + (y - k)² = r²

 

where:

(h, k) is the centerpoint

r is the radius

 

 

1)

 

x² - 8x + y² + 12y = -2

 

 

We need to complete the square.  Add 16 and add 36 on both sides of the equation.

 

x² - 8x + 16 + y² + 12y + 36 = -2 + 16 + 36

 

(x - 4)² + (y - 6)² = 50

 

 

The centerpoint is (4, 6).

The radius is 5√2.

 

 

To find the tangent, we take the derivative of the circle function using implicit differentiation.

 

2x + 2yy' - 8 + 12y' = 0

 

 

Isolate all the y' terms to one side of the equation.

 

2yy' + 12y' = -2x + 8

 

 

Factor out a y'.

 

y' (2y + 12) = -2x + 8

 

y' = (-2x + 8) / (2y + 12)

 

 

Evaluate the derivative at x=5 and y=1.

 

y' = (-10 + 8) / (2 + 12)

 

y' = -2 / 14

 

y' = -1 / 7

 

This is the slope of the tangent line.

 

 

Now we use the equation of a line and plug in the known values to solve for b.

 

y = mx + b

 

1 = (-1/7)(5) + b

 

1 = (-1 / 7) + b

 

8 / 7 = b

 

 

The equation of the tangent is

 

y = (-1 / 7)x + (8 / 7)

 

 

3)

 

Diameter = 2(5√2)

 

               = 10√2

 

 

When the chord coincides with the diameter, there is an intersection.

 

 

We can use the diameter and the centerpoint to find the endpoints of the diameter.  This can be achieved by using the distance formula.

 

10√2 = √[(x₁ - 4)² + (y₁ - 6)²]

 

 

Square both sides of the equation.

 

200 = (x₁ - 4)² + (y₁ - 6)²

 

200 = 100 + 100    --->  using a sum of squares

 

 

So

 

(x₁ - 4)² = 100         and           (y₁ - 6)² = 100

 

x₁ = 14                    and            y₁ = 16

 

 

This is an endpoint of the diameter.  Next, we can use these endpoints and the centerpoint to find the other endpoint of the diameter.

 

We can achieve this using the midpoint formula.

 

(4, 6) = ((14 + x₂) / 2  ,  (16 + y₂) / 2 )

 

 

Break down the equation by their points.

 

4 = (14 + x₂) / 2                    6 = (16 + y₂) / 2

 

8 = 14 + x₂                            12 = 16 + y₂

 

-6 = x₂                                    -4 = y₂

 

 

Now we find the slope of the diameter using the points.

 

m = (y₂ - y₁) / (x₂ - x₁)

 

m = (4 - 16) / (-6 - 14)

 

m = -12 / -20

 

m = 3 / 5

 

 

The slope of the chord will be perpendicular to the diameter.  It also goes through the point (5, 1).  Use the equation of a line and the slope to find the value of b.

 

y = mx + b

 

1 = (-5/3)(5) + b

 

1 = (-25 / 3) + b

 

28/3 = b

 

 

The equation of a chord is

 

y = (3 / 5)x + (28 / 3)

 

 

4)

 

 

To find an endpoint of the chord, we find the point of intersection of the chords equation and the circle equation.

 

x² + y² - 8x + 12y = -2                 eq1

 

y = (3 / 5)x + (28 / 3)                  eq2

 

 

 

Substitute eq2 into eq1 to get eq1 in terms of x.

 

x² + [(3/5)x + (28/3)]² - 8x + 12[(3/5)x + (28/3)] = -2

 

 

Solve for x from this equation.  Once you solve for x, substitute it into eq2 to solve for y.

 

 

 

I hope all of my guidance has helped you here.  I must say that this question was quite the challenge.