Two circles of radius 1 are externally tangent at Q . Let PQ and QR be diameters of the two circles. From P a tangent is drawn to the circle with diameter QR , and from R a parallel tangent is drawn to the circle with diameter PQ . Find the distance between these two tangent lines.
Could someone help me please, I tried drawing right triangles but they didn't seem to work....Thanks
Let O1 & O2 be the centres of the two circles. From P draw a tangent PT meeting the second circle at T. Likewise RS//PT is tangent to the first circle. From rt. triangle PTO2, sin(TPO2)= TO2/PO2 = 1/3. The distance between RS & PT = PR*sin(TPO2)= 4*1/3 = 4/3
Thanks a lot!!!
To solve this problem, we can break it down into smaller steps. Let's start by finding the length of segment PR.
Since the circles are externally tangent at Q, the centers of the circles, P and R, are joined by a straight line passing through Q. Since the radius of each circle is 1, we have two right triangles, PTQ and TRQ, where PT and TR are radii.
Using the Pythagorean theorem, we can find the length of segment PR.
Let's assume the coordinates of Q are (0,0), and the coordinates of R are (1,0). Since the circles are externally tangent, the coordinates of P will be (2,0).
Using the distance formula, we have:
PR = sqrt((2 - 1)^2 + (0 - 0)^2) = sqrt(1 + 0) = sqrt(1) = 1
So the length of segment PR is 1.
Now, let's find the length of the tangent lines drawn from P and R.
The tangent line drawn from P to the circle with diameter QR is perpendicular to segment PR. Therefore, we have a right triangle, PTR, where the length of segment PR is the hypotenuse.
Since PR = 1, the length of the tangent line drawn from P is the height of this right triangle. Let's call this length h.
Using the Pythagorean theorem, we can solve for h:
h^2 + 1^2 = PR^2
h^2 + 1 = 1^2
h^2 = 1 - 1
h^2 = 0
h = 0
So the length of the tangent line drawn from P is 0.
Now, let's find the distance between the two tangent lines.
Since the tangent line drawn from R is parallel to the tangent line drawn from P, the distance between them is equal to the length of the tangent line drawn from R.
Again, using the Pythagorean theorem, we can solve for the length of the tangent line drawn from R.
Since PR = 1, the length of the tangent line drawn from R is the height of the right triangle PRS, where PS is perpendicular to segment PR. Let's call this length d.
Using the Pythagorean theorem, we have:
d^2 + 1^2 = PR^2
d^2 + 1 = 1^2
d^2 = 1 - 1
d^2 = 0
d = 0
So the length of the tangent line drawn from R is 0.
Therefore, the distance between the two tangent lines is 0.
Sure, I can help you with that!
To find the distance between the two tangent lines, we can break it down into smaller steps:
Step 1: Draw the diagram
Start by drawing two circles with radius 1, externally tangent at point Q. Then draw diameters PQ and QR.
Step 2: Find the coordinates of the points P and R
Since the circles have radius 1, the distance from the center of each circle to the point of tangency (Q) is also 1. Therefore, the coordinates of point P can be (1, 0) and the coordinates of point R can be (-1, 0).
Step 3: Find the equation of the line passing through P and tangent to the circle with diameter QR
The radius QR is perpendicular to line PQ. Therefore, the line passing through P and tangent to the circle formed by diameter QR will be perpendicular to QR as well. Since QR is horizontal, the tangent line will have a slope of 0. So, the equation of the line passing through P can be written as y = k, where k is equal to the y-coordinate of P, which is 0.
Step 4: Find the equation of the line passing through R and parallel to the circle's tangent at Q
The slope of the tangent line at Q will be equal to the slope of RQ. The slope of RQ is given by (y2 - y1) / (x2 - x1), which is (0 - 1) / (0 - (-1)) = -1. Therefore, the equation of the line passing through R and parallel to the tangent line at Q can be written as y = mx + c, where m is -1 (the slope of RQ) and c is the y-coordinate of R, which is 0.
Step 5: Find the intersection point of the two lines
To find the intersection point of the two lines, set the equations of the lines equal to each other and solve for the x-coordinate:
0 = -x + c
By substituting 0 for y, we find x = c.
Step 6: Find the distance between the tangent lines
The distance between the two tangent lines is equal to the difference between their y-coordinates. In this case, their y-coordinates are 0. So, the distance between the tangent lines will also be 0.
Therefore, the distance between the two tangent lines is 0, meaning they are coincident lines.
I hope this explanation helps! Let me know if you have any further questions.