what is equation of circles passing through 4,2 and touching lines 4x-3y=5 and 3x+4y=10?

Answer 1

There are two circles satisfying the conditions.

The given lines are:
4x - 3y = 5
3x + 4y = 10

Find the point of intersection P, of the two lines. Solving two equations in two unknowns we have:
P(2, 1)

The center of circle will be on the bisector of the the angle formed by the two lines.

The slopes of the lines are:

tanα = 4/3
tanβ = -3/4

The slope of the bisecting line is tan[(α + β)/2]. Using the angle addition formula for tangents we have:

tan(α + β) = (tanα + tanβ) / [1 - (tanα)(tanβ)]
tan(α + β) = (4/3 - 3/4) / [1 - (4/3)(-3/4)]
tan(α + β) = (7/12) / 2 = 7/24

Now calculate:
sin(α + β) = 7/25
cos(α + β) = 24/25

Using the half angle formula for tangents we have:
tan[(α + β)/2] = [1 - cos(α + β)] / sin(α + β)
tan[(α + β)/2] = [1 - 24/25] / (7/25) = (1/25) / (7/25) = 1/7

With the slope and a point P, we can write the equation of the bisecting line.

y - 1 = (1/7)(x - 2)
7y - 7 = x - 2
x = 7y - 5

The distance of the center of the circle (x,y) to the point P(4,2) should be the same as the distance to both lines. But since the distance to both of the lines is the same, we only need to calculate the distance to one of the lines. Let's use the line:

4x - 3y = 5
4x - 3y - 5 = 0

The distance is the radius of the circle.

r² = (x - 4)² + (y - 2)² = (4x - 3y - 5)² / (4² + 3²)

Substitute x = 7y - 5 and solve for y.

(7y - 5 - 4)² + (y - 2)² = (4(7y - 5) - 3y - 5)² / (16 + 9)
(7y - 9)² + (y - 2)² = (28y - 20 - 3y - 5)² / 25
(7y - 9)² + (y - 2)² = (25y - 25)² / 25
(7y - 9)² + (y - 2)² = 25(y - 1)²

49y² - 126y + 81 + y² - 4y + 4 = 25y² - 50y + 25
50y² - 130y + 85 = 25y² - 50y + 25
25y² - 80y + 60 = 0
5y² - 16y + 12 = 0
(5y - 6)(y - 2) = 0
y = 6/5, 2

x = 7y - 5
x = 7(6/5) - 5 = 42/5 - 5 = 17/5
x = 7*2 - 5 = 14 - 5 = 9

The two centers of the circles are (17/5, 6/5) and (9, 2).

Now calculate the distance of each center to the line to get the radius of the circle.

d1 = | 4(17/5) - 3(6/5) - 5 | / √(4² + 3²)
d1 = | 68/5 - 18/5 - 5 | / 5 = 5/5 = 1

d2 = | 4*9 - 3*2 - 5 | / √(4² + 3²)
d2 = | 36 - 6 - 5 | / 5 = 25/5 = 5

The equations of the two circles are:

(x - 17/5)² + (y - 6/5)² = 1
(x - 9)² + (y - 2)² = 25