Known cycle c: (x+2)? +y？ =4, two mutually perpendicular straight lines L 1, L2 both pass through point A (a, 0).

(1) If both L 1 and L2 are tangent to circle C, find the equation of straight line L 1 and L2.

(2) When a=2, if the circle with the center M( 1, m) and the circle c are tangent to the straight line L 1, L2, then the equation of the circle m is found.

(3) When a=- 1, find the maximum sum of the chord lengths of L 1 and L2 cut by circle C. ..

Solution: (1) According to the meaning of the question, the center of the circle is c (-2,0), and on the X axis, the radius is R = 2；;

The intersection of two vertical straight lines L 1 and L2 is A(a, 0), which is also on the X axis.

Then drawing shows: AC=√2r, that is |a+2|=2√2, and the solution is A =-2 2√2； √ 2;

As can be seen from the figure, two vertical straight lines have two positions,

The slope of a straight line is tan 45 = 1, or tan135 =-1;

Then the equations of two vertical straight lines L 1 and L2 are:

Y=x+2-2√2，y =-x-2+2√2；

Or y=x+2+2√2 and y =-x-2-2 √ 2;

(2) Let the equation of circle M be (x- 1)? +(y-m)？ =r？ , and then:

MA =√2r； MC = R+R； Namely √(m? + 1)=√2r； √(m？ +9)= 2+r；

Simultaneous solution: m=√7, r = 2；;

So the equation of circle m is: (x- 1)? +(y-√7)？ =2? ;

(3) The problem can be simplified as the following model:

Two vertical straight lines starting from the point with the center of 1 and passing through a circle with a radius of 2.

What is the maximum value of the sum d of two chord lengths cut by a circle?

Let the distance between the center of the circle and two vertical straight lines be m and n, then: m? +n？ = 1;

So m? +n？ ≥2mn， 1≥2mn，m？ n？ ≤ 1/4;

And this model is easy to know: d=2√(4-m? )+2√(4-n？ );

Square two sides: d? =4×[4-m？ +4-n？ +2√((4-m？ )(4-n？ ))]

d？ =4×[8-(m？ +n？ )+2√( 16-4(m？ +n？ )+m？ n？ )]

=4×[8- 1+2√( 16-4+m？ n？ )]

=4×[7+2√( 12+m？ n？ )]

≤4×[7+2√( 12+( 1/4))]=56;

That is, d≤2√ 14 and dmax=2√ 14.