Let the equation of the circle is `x^2 + y^2 = 4.` Find the total number of points on `y = |x|` from which perpendicular tangents can be drawn.

Answer

To solve the problem, we need to find the total number of points on the line \( y = |x| \) from which perpendicular tangents can be drawn to the circle defined by the equation \( x^2 + y^2 = 4 \).

### Step 1: Understand the Circle
The equation of the circle is given as:
\[ x^2 + y^2 = 4 \]
This represents a circle centered at the origin \((0, 0)\) with a radius of \( r = 2 \) (since \( r^2 = 4 \)).

### Step 2: Understand the Line
The line \( y = |x| \) consists of two parts:
1. \( y = x \) for \( x \geq 0 \)
2. \( y = -x \) for \( x < 0 \)

### Step 3: Find the Condition for Perpendicular Tangents
For a point \( P(a, |a|) \) on the line \( y = |x| \), we need to find the points from which perpendicular tangents can be drawn to the circle. The condition for perpendicular tangents to be drawn from an external point \( P(a, |a|) \) to the circle is that the distance from the point to the center of the circle must be greater than the radius.

The distance \( d \) from the point \( P(a, |a|) \) to the center \( (0, 0) \) is given by:
\[ d = \sqrt{a^2 + |a|^2} \]
Since \( |a|^2 = a^2 \) for \( a \geq 0 \) and \( |a|^2 = a^2 \) for \( a < 0 \), we can simplify this to:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = |a|\sqrt{2} \]

### Step 4: Set Up the Inequality
For perpendicular tangents to exist, we need:
\[ d > r \]
Substituting the values we have:
\[ |a|\sqrt{2} > 2 \]
Dividing both sides by \( \sqrt{2} \):
\[ |a| > \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 5: Determine the Points on the Line
The points on the line \( y = |x| \) that satisfy \( |a| > \sqrt{2} \) are:
1. For \( y = x \): \( x > \sqrt{2} \)
2. For \( y = -x \): \( x < -\sqrt{2} \)

### Step 6: Count the Points
The points on the line \( y = |x| \) can be represented as:
- For \( x > \sqrt{2} \): There are infinitely many points.
- For \( x < -\sqrt{2} \): There are also infinitely many points.

Since both sections of the line yield infinitely many points, we conclude that there are infinitely many points on the line \( y = |x| \) from which perpendicular tangents can be drawn to the circle.

### Final Answer
The total number of points on \( y = |x| \) from which perpendicular tangents can be drawn to the circle is **infinite**.