From the graph below, there are obviously two solutions:
We set the two green radii, (or the two red radii, equal), and also equate
that to the perpendicular distance from (h,k) to the line 5x - y - 17 = 0
Set the first two equal, simplify, and square both sides.
Set the second and third equal, simplify, and square both sides.
Solve for one of the letters in one equation. Substitute it
in the other.
You'll get two solutions for (h,k), and then substitute in one of the
expressions above for the two radii. Then substitute in the standard
formula for a circle:
to get the two equations.
If you need help solving that, tell me in the thank-you note form
below and I'll get back to you by email to help you. No charge ever!
Edwin
.
Write the equations of the circles satisfying the condition, tangent to 5x - y - 17 = 0 at (4,3) and also tangent to x - 5y - 5 = 0
~~~~~~~~~~~~~~~~~~~~~~
The key step to solve this problem is to find the center of the circle.
From one side, the center lies on the perpendicular to the first line passing through the given point.
From the other side, the center lies on the angle bisector of the angle formed by the two given lines
(since this angle bisector is the locus of points equidistant from the lines - very elementary basic property of the angle bisector . . . )
So, our goal is to find the intersection point of the perpendicular and the angle bisector.
Since we have, actually, TWO different adjacent angles between the given lines, we will have TWO different angle bisectors that would create
two different intersection points and, correspondingly, two different circles.
OK. So we just have an idea on what we want and how to do it. Now let's implement it.
The straight line 5x - y - 17 = 0 has the slope m = 5. Hence, the perpendicular line has the slope = -0.2.
The equation of the line having the slope -0.2 and passing through the point (4,3) is
y - 3 = -0.2*(x-4), or, which is the same, x + 5y = 19. (1)
Thus the perpendicular to the first given line at the given point is (1).
So, the first part of the work is done. We got the equation of the perpendicular line.
Next goal is to write an equation (equations) for the angle bisector/bisectors to the two given lines.
Probably, you never did it at school, but I will teach you right now on how to do it.
The distance from the point (x,y) in a coordinate plane to the line 5x - y - 17 = 0 is = .
See the lesson The distance from a point to a straight line in a coordinate plane in this site.
Similarly, the distance from the point (x,y) in a coordinate plane to the line x - 5y - 5 = 0 is = .
If the point (x,y) lies on the angle bisector between the given lines, then the point (x,y) is equidistant from the lines,
which gives you an equation
= , or, canceling in both sides, equivalently,
= . (2)
From learning absolute values in the school, you must know that the equation (2) is equivalent to the SET of these two equations
5x - y - 17 = x - 5y -5, (3) and
5x - y - 17 = -(x - 5y -5). (4)
Next, simplifying, you have
4x + 4y = 12 (5) instead of (3), and
6x - 6y = 22 (6) instead of (4).
Actually, equation (5) is for one angle bisector, while equation (6) is for another angle bisector.
By the way, notice in your mind that the straight lines (5) and (6) are perpendicular,
which you may expect for angle bisectors of the two adjacent supplementary angles.
Now we are at the finish line. We must solve the equation (1) with the equation (5) to get one intersection point (=the center),
and we must solve the equation (1) with the equation (6) to get the second intersection point (=the center of the second circle).
Solving equations (1) and (5) together
x + 5y = 19,
4x + 4y = 12,
you get the solution x = -1, y = 4. It is your first center.
Solving equations (1) and (6) together
x + 5y = 19,
6x - 6y = 22,
you get the solution x = , y = . It is your second center (of the second circle).
Having the centers, you can easily calculate the radii of the circles as distances to the given point.
Since Edwin just gave the plot in his post, I will not repeat it again.