Problem: Find the radius of the largest circle laying in the first quadrant and touching 4x, 3y=12and the coordinate axes.

Solution:

Step 1: Plot the given lines and coordinate axes on the graph.



Step 2: Find the distance between the origin and the line 4x.

The distance between the origin and the line 4x is given by the formula:

d = |ax + by + c| / √(a² + b²)

where,

a = coefficient of x in the equation of the line
b = coefficient of y in the equation of the line
c = constant term in the equation of the line

In this case, the equation of the line is 4x = 0, which can be written as 4x - 0y = 0. So, a = 4, b = 0, and c = 0. Substituting these values in the formula, we get:

d = |4(0) + 0(0) + 0| / √(4² + 0²) = 0

So, the distance between the origin and the line 4x is 0.

Step 3: Find the distance between the origin and the line 3y=12.

The distance between the origin and the line 3y = 12 is given by the formula:

d = |ax + by + c| / √(a² + b²)

In this case, the equation of the line is 0x + 3y = 12, which can be written as 0x + 3y - 12 = 0. So, a = 0, b = 3, and c = -12. Substituting these values in the formula, we get:

d = |0(0) + 3(0) - 12| / √(0² + 3²) = 4

So, the distance between the origin and the line 3y = 12 is 4.

Step 4: Find the distance between the origin and the point of intersection of the lines 4x and 3y = 12.

To find the point of intersection of the lines 4x and 3y = 12, we can solve the two equations simultaneously.

4x = 0 and 3y = 12

So, x = 0 and y = 4. Therefore, the point of intersection is (0, 4).

The distance between the origin and the point (0, 4) is given by the formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

where,

(x₁, y₁) = coordinates of the origin
(x₂, y₂) = coordinates of the point (0, 4)

Substituting the values, we get:

d = √((0 - 0)² + (4 - 0)²) = 4

So, the distance between the origin and the point of intersection of the lines 4x and 3y =