p:q is the sub duplicate ratio of (p-x^2):(q-x^2) . Then x^2 is
A . p/(p+q)
B . q/(p+q)
C . pq/(p+q)
D . None of the above
Please answer I tried but I didn't got answer
I actually had to look this up, had never heard of it.
Subduplicate ratio: The sub duplicate ratio m : n is the ratio √m : √n. So, the sub duplicate ratio of the ratio m22 : n22 is the ratio m : n.
For example:
The sub duplicate ratio of 25 : 81 = √25 : √81 = 5 : 9
so let's just go by that definition:
given:
the sub duplicate ratio of (p-x^2):(q-x^2) = p:q
then
the sub duplicate ratio of √(p-x^2):√(q-x^2) = p:q
square both sides:
(p-x^2)/(q-x^2) = p^2/q^2
pq^2 - q^2 x = qp^2 - p^2x
p^2x - q^2x = qp^2 - pq^2
x(p^2 - q^2) = pq(p - q)
x = pq(p - q)( (p+q)(p-q) )
= pq/(p+q) , where p ≠ q
looks like C
To find the value of x^2, let's analyze the given information.
According to the problem, the sub duplicate ratio of (p - x^2) : (q - x^2) is given as p : q.
Subduplicate ratio means that the square root of the first term divided by the square root of the second term is equal to the ratio given.
Using this information, we can set up the following equation:
√(p - x^2) / √(q - x^2) = p / q
Now, let's simplify this equation step-by-step:
1. Cross multiply to eliminate the square roots:
√(p - x^2) * q = √(q - x^2) * p
2. Square both sides to eliminate the square roots:
(√(p - x^2) * q)^2 = (√(q - x^2) * p)^2
Expanding both sides:
(p - x^2) * q^2 = (q - x^2) * p^2
3. Expand further:
pq^2 - x^2q^2 = pq^2 - x^2p^2
4. Cancel out the common terms on both sides:
pq^2 - x^2q^2 = pq^2 - x^2p^2
- x^2q^2 = - x^2p^2
Since the common terms cancel out, we can conclude that q^2 = p^2.
Taking the square root of both sides:
√(q^2) = √(p^2)
q = p
Now, we need to find x^2. Let's substitute q = p back into the original equation:
(p - x^2) : (p - x^2) = p : p
As we simplified the equation, we can see that both sides are equal. Therefore, the value of x^2 can be any real number, and it cannot be determined from the given information.
Hence, the answer is D. None of the above.
To find the value of x^2, we can set up a proportion using the given information:
The sub duplicate ratio of (p-x^2):(q-x^2) is given as p:q.
Using the proportion formula, we can write:
(p - x^2) / (q - x^2) = p / q
Cross-multiplying, we get:
q * (p - x^2) = p * (q - x^2)
Expanding the equation, we have:
pq - qx^2 = pq - px^2
Simplifying, we find:
-qx^2 = -px^2
Dividing both sides by -x^2 (since x^2 ≠ 0), we get:
q = p
So, if p:q is the sub duplicate ratio of (p-x^2):(q-x^2), then p and q must be equal. Therefore, the value of x^2 does not affect the ratio, and we cannot determine its value based on the given information.
Hence, the answer is D. None of the above.