SOLUTION: Find the equation of the tangents drawn from the point (4, 7) to the circle: (x - 2)² + y² + 4y = 0.

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Question 1209260: Find the equation of the tangents drawn from the point (4, 7) to the circle:
(x - 2)² + y² + 4y = 0.

Found 4 solutions by mccravyedwin, math_tutor2020, Edwin McCravy, AnlytcPhil:
Answer by mccravyedwin(405) About Me  (Show Source):
You can put this solution on YOUR website!

%28x-2%29%5E2%2By%5E2%2B4y=0

Complete the square on the y-terms by adding 4 to both sides.

%28x-2%29%5E2%2By%5E2%2B4y%2B4=4

Factor the trinomial:

%28x-2%29%5E2%2B%28y%2B2%29%5E2=4

Compare to the standard form of a circle:

%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2

And we see that the center of the circle is (h,k) = (2,-2)
and the radius is r=2

So we sketch the figure:



Let X and Q be the points of tangency, we draw radii OX and OQ,
then we draw OP and draw XY parallel to the x-axis.

We want the equations of PX and PQ.

We now see that the equation of the tangent PQ 
is the vertical line x = 4. 

We also see that PQ is 9 units in length, 7 units above the x-axis
and 2 units below the x-axis.

From right triangle OPQ, we see that

tan%28%22%22%3COPQ%29=OQ%2F%28PQ%29=2%2F9

%22%22%3CXPQ=2%2A%22%22%3COPQ

We use the identity tan%282theta%29%22%22=%22%22%282tan%5E%22%22%28theta%29%29%2F%281-tan%5E2%28theta%29%29

 tan%28%22%22%3CXPQ%29%22%22=%22%22%282%2Atan%5E%22%22%28%22%22%3COPQ%29%29%2F%281-tan%5E2%28%22%22%3COPQ%29%29%22%22=%22%22%282%2A%282%2F9%29%5E%22%22%29%2F%281-%282%2F9%29%5E2%29%22%22=%22%22%284%2F9%29%2F%281-4%2F81%29%22%22=%22%22%284%2F9%29%2F%2877%2F81%29%22%22=%22%22%284%2F9%29%2881%2F77%29%22%22=%22%2236%2F77

Since %22%22%3CXPQ%22%22=%22%22%22%22%3CXPY, by right triangle PXY,

%22%22%3CPXY and %22%22%3CXPY are complementary and
matrix%281%2C3%2Cslope%2Cof%2CPX%29%22%22=%22%22tan%28%22%22%3CPXY%29%22%22=%22%22 cot%28%22%22%3CXPY%29%22%22=%22%221%5E%22%22%2Ftan%5E%22%22%28%22%22%3CXPY%29%22%22=%22%221%5E%22%22%2Fexpr%2836%2F77%29%5E%22%22%22%22=%22%2277%2F36

Finally we use the point-slope formula to find the equation of tangent PX

y-y%5B1%5D%22%22=%22%22m%2A%28x-x%5B1%5D%29

y-7%22%22=%22%22expr%2877%2F36%29%28x-4%29

y-7%22%22=%22%22expr%2877%2F36%29x-77%2F9%29

y%22%22=%22%22expr%2877%2F36%29x-77%2F9%2B7%29

y%22%22=%22%22expr%2877%2F36%29x-77%2F9%2B63%2F9%29

y%22%22=%22%22expr%2877%2F36%29x-14%2F9%29

That's the equation of the tangent PX

and the equation of the tangent PQ is the vertical line x = 4.

Edwin

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

I'll provide a rough outline.
The scratch work will be left for the student to do.

Let (4,7) be the location of point A.

(x-2)^2 + y^2 + 4y = 0
turns into
(x-2)^2 + (y+2)^2 = 4
after completing the square

Compare this to the circle template (x-h)^2+(y-k)^2 = r^2
point B = (h,k) = (2,-2) = center of the circle
r^2 = 4 ---> r = 2 is the radius

C = midpoint of A and B
Use the midpoint formula, or follow a process similar to this question, to find that C = (3, 2.5)

Then use the distance formula
d+=+sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29
to find out how far it is from A to B.

You should determine that segment AB is exactly sqrt%2885%29 units long.

This is then split in half to get AC+=+BC+=+sqrt%2885%29%2F2
This is the radius of a larger circle centered at point C
The equation of this larger circle is %28x-3%29%5E2+%2B+%28y-2.5%29%5E2+=+21.25 where the 21.25 is the result of computing r%5E2+=+%28sqrt%2885%29%2F2%29%5E2

Gather up the equations of each circle to form this system of equations.

Solving that system generates these intersection points
D = (16/85, -98/85)
E = (4,-2)
These are the points of tangency.
You can use Thales Theorem to prove this.
Thales Theorem is a special case of the Inscribed Angle Theorem.

16/85 = 0.188235294118 approximately
-98/85 = -1.152941176471 approximately

--------------------------------------------------------------------------

Focus on these points
A = (4,7)
D = (16/85, -98/85)
E = (4,-2)

The equation of tangent line AD is y = (77/36)x - 14/9
77/36 = 2.1388889 approximately where the 8s go on forever but we have to round at some point
14/9 = 1.55556 where the 5s go on forever but we have to round at some point

The equation of tangent line AE is x = 4
This is a vertical line through 4 on the x axis.

You can use a tool like GeoGebra to verify the answers.

Here is the link to the GeoGebra worksheet
https://www.geogebra.org/calculator/f6yrhtke

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
You could probably also find the slanted tangent line by 

the equation of the circle and the point (4,7) with

y = mx + b

and finding the values of b and m that would produce
a double root, by setting the discriminant = 0.

Edwin

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!

Yes it can be done that way:

We use the point-slope formula for the equation of a line through (4,7) 

y-7=m%28x-4%29
y=m%28x-4%29%2B7

Substitute in

%28x+-+2%29%5E2+%2B+y%5E2+%2B+4y+=+0

%28x+-+2%29%5E2+%2B+%28m%28x-4%29%2B7%29%5E2+%2B+4%28m%28x-4%29%2B7%29+=+0

That simplifies to the quadratic in x2

+%28m%5E2%2B1%29x%5E2%2B%28-8m%5E2%2B18m-4%29x+%2B+16m%5E2-72m%2B81=0

The point of tangency can be thought of as where 
the tangent line intersects the circle TWICE at
the same point.  So both solutions must be equal.

So we set the discriminant+=+b%5E2-4ac+=+0 

%28-8m%5E2%2B18m-4%29%5E2-4%28m%5E2%2B1%29%2816m%5E2-72m%2B81%29=0

That simplifies to 

144m-308=0

m=308%2F144=77%2F36

The equation of the slanted tangent line is

y=m%28x-4%29%2B7

y=expr%2877%2F36%29%28x-4%29%2B7

Which simplifies to what I got above.

Edwin