How to find point of intersection of direct common tangents for two circles which touch each other?
To find the point of intersection of direct common tangents for two circles that touch each other, you can follow these steps:
Step 1: Understand the problem
Visualize the two circles that touch each other externally. To find the point of intersection of the direct common tangents, we need to determine the tangent points on each circle and connect them.
Step 2: Find the equation of each circle
To find the tangent points, we first need to know the equation of each circle. Let's assume the first circle has center C1(x1, y1) and radius r1, and the second circle has center C2(x2, y2) and radius r2. The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Step 3: Find the tangent points
The tangents will be perpendicular to the line joining the centers of the circles.
First, find the slope of the line joining the centers of the circles:
m = (y2 - y1) / (x2 - x1)
The slope of the tangent will be the negative reciprocal of this slope:
m_tangent = -1 / m
Next, find the midpoint M of the line joining the centers:
M(xm, ym) = ((x1 + x2) / 2, (y1 + y2) / 2)
Step 4: Find the tangent points on each circle
Using the midpoint of the line joining the centers (M) and the slope of the tangent (m_tangent), we can find the tangent point for each circle.
To find the tangent point on the first circle, substitute the values of C1(x1, y1), r1, M(xm, ym), and m_tangent into the following equations:
(x - x1)^2 + (y - y1)^2 = r1^2
y - ym = m_tangent * (x - xm)
Solve these two equations simultaneously to find the values of x and y for the tangent point on the first circle.
Similarly, find the tangent point on the second circle using the values of C2(x2, y2), r2, M(xm, ym), and m_tangent.
Step 5: Draw the tangents and find the point of intersection
Connect the tangent points on each circle with a line. The point where these lines intersect will be the point of intersection of the direct common tangents.
By following these steps, you should be able to find the point of intersection of direct common tangents for two circles that touch each other.