Mathematicians, I need help please read the math problem below.?

Radius of the circle
= Distance between the center and the tangent
= |3(-4) + 4(-2) - 5)| / √[(-4)² + (-2)²]
= 25 / √20
= 25 / 2√5
= (5√5)/2

The distance from the point (x₀,y₀) to the line ax+by+c=0 is |ax₀ + by₀ + c| / √(a² + b²)
This radius is |3(-4) + 4(-2) - 5| / √(3² + 4²) which simplifies to 5.

The line perpendicular to ax+by+c through a given point (x₀,y₀) is bx-ay-(bx₀-ay₀)=0.
The equation of the diameter perpendicular to 3x+4y-5=0 is 4x-3y-(4(-4)-3(-2))=0 which simplifies to 4x-3y-10=0

Graph: https://www.desmos.com/calculator/rvwuqvds6o

center = (-4, -2)
tangent to 3x + 4y - 5 = 0

let the point of tangency be (x,y)

This satisfies the equation of the line.
3x + 4y - 5 = 0
3x + 4y = 5
4y = - 3x + 5
y = -3/4x + 5/4

The slope m of the radius from (-4,-2) to (x,y) is

m = (y--2)/(x--4) = (-3/4x+5/4+2)/(x+4)
= (-3/4x + 13/4)/(x+4)

The radius is perpendicular to the line The slope of the radius is -1/(-3/4) = 4/3.

4/3 = (-3/4x + 13/4)/(x+4)
4/3(x+4) = -3/4x + 13/4
x+4 = (3/4)(-3/4x + 13/4)
x+4 = -9/16x + 39/16
25/16 x = 39/16 - 64/16 = -25/16
x = -1

(x,y) is on both lines
y = -3/4x + 5/4
y = -3/4(-1) + 5/4
y = 3/4+5/4 = 8/4 = 2
(x,y) = (-1,2)

The length of the radius is
r² = (x--4)² + (y--2)²
= (-1--4)² + (2--2)²
= (4-1)² + (2+2)²
= 3² + 4² = 9 + 16 = 25
r = 5

radius is 5

Michael used the formula for the distance between a point (x,y) and a line {ax + by + c = 0}

d = |ax + by + c| / √(a² + b²)
(-4,2) {3x + 4y - 5 = 0}

d = |3(-4) + 4(-2) - 5| / √(3²+4²)
= | -12 -8 -5| / 5 = |-25|/5 = 25/5 = 5

Equation of the circle
(x+4)^2+(y+2)^2 = r^2
2(x+4) + 2(y+2) dy/dx = 0
2(y+2) dy/dx = -2(x+4)
dy/dx = -(x+4)/(y+2)

3x+4y=5 (tangent line)
4y = -3x+5
y = (-3/4) x + 5/4
slope = -3/4


dy/dx = -3/4
-(x+4)/(y+2) = -3/4
4(x+4)= 3(y+2)
4x+16 = 3y+6
3y-4x = 10 ----(1)
4y+3x= 5 -----(2)

solving x=-1 , y=2
(x+4)^2+(y+2)^2 = r^2
(-1+4)^2 +(2+2)^2 = r^2
9+16 = r^2
r = 5

Well,

the radius of the circle is therefore the distance
between centre C(-4, -2) to its perpendicular projection on the line
hence, the distance from C to the line is given by the formula :
R = | 3*(-4) + 4*(-2) - 5 | / sqrt( 3^2 + 4^2 )
= | -12 - 8 - 5 | / 5
= 25/5
= 5

the radius of the circle is : R = 5

hope it' ll help !!