If a, b ( b is greaterthan a) be the diameters of two concentric circles and 'c' be the length of a chord of a circle which is tangent to the other circle, then find the value of b in terms of a and c.
did you make a sketch?
Draw a common diameter.
The radius of the larger circle will be b/2, and that of the smaller circle is a/2
draw the tangent so that it hits the end of the diameter.
you will have a right-angled triangle, so that
(b/2)^2 = (c/2)^2 + (a/2)^2
b^2 /4 = c^2/4 + a^2/4
b^2 = c^2 + a^2 <----- that should do it
or
b = √(c^2 + a^2)
To find the value of b in terms of a and c, we need to make use of the properties of tangent and chord in a circle. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw two concentric circles with center O, where a and b are the diameters of the two circles. Draw a chord CD in the outer circle, such that it is tangent to the inner circle at point T.
O
/ \
/ \
/ \
A-------B
| |
C----T--D
Step 2: Understanding the properties
In a circle, when a chord intersects a tangent, the product of the lengths of the segments of the chord is equal to the square of the length of the tangent segment.
Using this property, we can write the equation:
AC * CD = TC^2
Step 3: Expressing AC and CD in terms of a and c
Since the line CD is a chord of the outer circle, its length is given as c.
AC is the radius of the inner circle, which is equal to half of the diameter a. Hence, AC = a/2.
Substituting these values, we have:
(a/2) * c = TC^2
Step 4: Expressing TC in terms of a and b
To express TC in terms of a and b, we need to find the relationship between the two. We know that OT is the radius of the outer circle, which is equal to b/2.
By subtracting TC from OT, we get:
OT - TC = b/2 - TC = (b - 2TC)/2
Step 5: Solving for b
Substituting the value of TC^2 from step 3, we have:
(a/2) * c = (b - 2TC)^2/4
Cross-multiplying, we get:
4(a/2) * c = (b - 2TC)^2
Expanding the equation, we have:
2ac = b^2 - 4bTC + 4TC^2
Rearranging, we get:
2ac = b^2 - 4bTC + 4TC^2
Simplifying further:
b^2 - 4bTC + 4TC^2 - 2ac = 0
This is a quadratic equation in b. By solving this equation using the quadratic formula, we can find the value of b in terms of a and c:
b = [4TC ± sqrt((4TC)^2 - 4*4TC^2 + 8ac)] / 2
Simplifying,
b = [4TC ± sqrt(16TC^2 - 32TC^2 + 8ac)] / 2
b = [4TC ± sqrt(-16TC^2 + 8ac)] / 2
Since we are given that b > a, we can take the positive square root and simplify further:
b = [4TC + sqrt(-16TC^2 + 8ac)] / 2
b = 2TC + sqrt(-16TC^2 + 8ac)
Therefore, the value of b in terms of a and c is:
b = 2TC + sqrt(-16TC^2 + 8ac)