Question

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

Answer 1

A(-2, 2) and B(6, 10)

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).