maths qn help (1 Viewer)

[Masaken]

"Find the equations of the four circles which are tangent to the x-axis, the y-axis and the line x + y = 2."

I've drawn out the circles and I know for each circle the distances from the centre to the respective points touching each mentioned line are equal because of radii, and I know that perpendicular distance is involved, but I don't know how to apply perpendicular distance or where to go from here. Any help on where to go after this would be great help, thanks!

 

[jimmysmith560]

Would the following working help?

Find the equations of the two circles in the first quadrant. Since the x-axis and y-axis are both tangent to the circle, the x-coordinate of the centre is equivalent to the x-intercept of the circle, and the y-coordinate of the centre is equivalent to the y-intercept of the circle. The distance from the centre of the circle to the x-intercept or y-intercept is the radius. This means that the x-intercept is (r,0), the y-intercept is (0,r) and the centre of the circle is (r,r), where is the radius of the circle.

So, this means the equation of the circle is . Now the distance from the line to the point (r,r) is a radius of the circle. To calculate this distance, we can use the perpendicular distance formula as the line from the centre of a circle to a tangent is perpendicular to that tangent. Sub in the line and the point (r,r), and let the perpendicular distance be equal to , since this distance is a radius. Solving for , you should get . Now sub these values into the equation of the circles to get two circles which satisfy the conditions.

 

[Masaken]

Would the following working help?

Find the equations of the two circles in the first quadrant. Since the x-axis and y-axis are both tangent to the circle, the x-coordinate of the centre is equivalent to the x-intercept of the circle, and the y-coordinate of the centre is equivalent to the y-intercept of the circle. The distance from the centre of the circle to the x-intercept or y-intercept is the radius. This means that the x-intercept is (r,0), the y-intercept is (0,r) and the centre of the circle is (r,r), where is the radius of the circle.

So, this means the equation of the circle is . Now the distance from the line to the point (r,r) is a radius of the circle. To calculate this distance, we can use the perpendicular distance formula as the line from the centre of a circle to a tangent is perpendicular to that tangent. Sub in the line and the point (r,r), and let the perpendicular distance be equal to , since this distance is a radius. Solving for , you should get . Now sub these values into the equation of the circles to get two circles which satisfy the conditions.

Sorry, I'm still a little confused... how did you end up getting those two values for r, and for the other two circles (the one in the second quadrant and the one in the fourth) where would you go with those?

 

[jimmysmith560]

Sorry, I'm still a little confused... how did you end up getting those two values for r, and for the other two circles (the one in the second quadrant and the one in the fourth) where would you go with those?

Inputting the values into the perpendicular distance formula will give you the following:



You could then solve in a way that would lead you to , resulting in two possibilities. By solving each equation, you should reach the following results:



Those values are essentially after they have been rationalised.

Subsequently, to get the other two circles, you need to make the centre (-r,r) and (r,-r). Since the line only passes through the 1st, 2nd and 4th quadrants, the circle can only be a tangent to the line in those quadrants.

The following working may be helpful in that regard:

 

[Masaken]

Oh, I get it now, thank you!