Question

The line y = x − 6 y=x−6 has exactly one point in common with the circle x 2 + y 2 = r 2 x2+y2=r2. what is the value of r 2 r2?

Answer 1

Given:
y = x - 6         (1)
 for the line
x² + y² = r²     (2)
 for the circle

When the line intercepts the circle, the x and y values will be equal.
Equate the values of y by substituting (1) into (2).
x² + (x - 6)² = r²
x² + x² - 12x + 36 = r²
2x² - 12x + 36 = r²

Answer:  r² = 2x² - 12x + 36