show that a.b.c.d are in proportion if ma2+nb2 : mc2 + nd2 : : ma2- nb2 : mc2 -nd2
Answers
Answer:
ab=cd
A2+b2=C2+D2
ad=bc
Yes, a.b.c.d are in proportion.
Given,
ma²+nb² : mc²+nd² :: ma²-nb² : mc²-nd²
To Find,
The a.b.c.d are in proportion.
Solution,
Before solving the question, we must understand the concept of extreme and mean in the proportions.
Extreme:- If a proportion is written in ratio form using a colon, the extremes are the values that are furthest apart.
For example:- p : q :: r : s. In this, 'p' and 's' is the extreme of proportions.
Mean:- In proportions, the means are the values associated with the middle terms.
For example:- p : q :: r : s. In this, 'q' and 'r' is the mean of proportions.
The formula of Proportions:
Product of extreme = product of mean.
⇒ p × s = q × r.
Now, let's solve the question,
This means that they are in proportion, so the product of extreme should be equal to the product of mean.
So, they can be written as
(ma² + nb²)/(mc² + nd²) = (ma² - nb²)/ ( mc² - nd²)
Let us assume that a.b.c.d are in proportion, so
a/b = c/d
Squaring on both sides
a²/b² = c²/d²
Multiplying both sides by m/n
ma²/nb² = mc²/ nd²
Applying the rule of adding the numerator and denominator and dividing by subtraction of the numerator and denominator on both sides,
(ma² + nb²)/ (ma² - nb²) = (mc² + nd²) / ( mc² - nd²)
Interchanging fractions,
(ma² + nb²)/(mc² + nd²) = (ma² - nb²)/ ( mc² - nd²)
So, we have arrived at this equation, and it is given that this is true, it can be concluded that our assumption was true.
Yes, a.b.c.d are in proportion.
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