Steps to solve the problem:

The first step is to transform or analyze the related formula by using the geometric meaning of the given formula;

The second step is to solve the problem by using the mathematical thought of mathematical combination and transformation;

The third step is to draw a conclusion.

The real number is known and the equation is satisfied.

Find the maximum and minimum values of (1);

(2) the minimum value;

(3) the maximum and minimum values.

solve

(1) As shown in the figure, the equation represents a point as the center and a circle as the radius.

Suppose, that is,

Then when the distance from the center of the circle to the straight line is the radius, the straight line is tangent to the circle, and the slope gets the maximum and minimum.

come from

, .

(2) If, then,

Only when the straight line and the circle are tangent to the fourth image, the intercept is the smallest.

According to the distance formula from point to straight line,

That is, therefore,

(3) is the square of the distance between the point on the circle and the origin,

So connect, intersect the circle at a point, and extend the intersection point at the point,

rule

,

.

Summarizing the transformation of related formulas or solving problems by using the geometric meaning of given formulas fully embodies the mathematical thought of the combination of numbers and shapes, among which the following transformations are very common, so we should pay attention to memory:

The maximum problem of (1) can be transformed into the maximum problem of the slope of the moving straight line.

(2) The maximum problem of shape can be transformed into the maximum problem of moving line intercept;

(3) The shape maximum problem can be transformed into the square maximum problem of the distance between two points.