A line and a circle intersect at the points A and B. Use the equations below to find the coordinates of the points of intersection A and B. y = x + 4 ( x − 3 ) 2 + ( y − 5 ) 2 = 34

Step-by-step explanation:

Given the equation of a line y = x + 4 and equation of a circle as

( x − 3 )² + ( y − 5 )² = 34, if the line and the circle intersect at points A and B, to get this points, we will substitute the equation of the line into that of the circle as shown;

We will have to expand the equation of the circle first before making the substitute.

( x − 3 )² + ( y − 5 )² = 34

x²-6x+9+y²-10y+25 = 34

x²+y²-6x+-10y+34-34 = 0

x²+y²-6x+-10y = 0

Substituting y = x+ 4 into the resulting expression;

x²+(x+4)²-6x+-10y = 0

x²+x²+8x+16-6x+-10(x+4) = 0

x²+x²+8x+16-6x+-10x-40 = 0

2x²-8x-24 = 0

x²-4x-12 = 0

(x²-6x)+(2x-12) = 0

x(x-6)+2(x-6) = 0

x+2 = 0 and x-6 = 0

x = -2 and 6

when x = -2;

y = -2+4

y = 2

when x = 6

y = 6+4

y = 10

The coordinates of the point of intersection are A(-2, 2) and B(6, 10).

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.