Given x = a 2 + b 2 + a 2 - b 2 a 2 + b 2 - a 2 - b 2 , use componendo and dividendo to prove that b 2 = 2 a 2 x x 2 + 1 .

Given: x = a 2 + b 2 + a 2 - b 2 a 2 + b 2 - a 2 - b 2 On using compendo and dividendo rule for the given, we get ⇒ x + 1 x - 1 = 2 a 2 + b 2 2 a 2 - b 2 Squaring on both the sides, we get ⇒ ( x + 1 ) 2 x - 1 2 = a 2 + b 2 2 a 2 - b 2 2 ⇒ x 2 + 2 x + 1 x 2 - 2 x + 1 = a 2 + b 2 a 2 - b 2 ⇒ x 2 + 2 x + 1 + x 2 - 2 x + 1 x 2 + 2 x + 1 - x 2 - 2 x + 1 = a 2 + b 2 + a 2 - b 2 a 2 + b 2 - a 2 - b 2 ⇒ b 2 = 4 a 2 x 2 x 2 + 1 ⇒ b 2 = 2 a 2 x x 2 + 1 Hence, it is proved.

Given x = a 2 + b 2 + a 2 - b 2 a 2 + b 2 - a 2 - b 2 , use componendo and dividendo to prove that b 2 = 2 a 2 x x 2 + 1 .

Given: x = a 2 + b 2 + a 2 - b 2 a 2 + b 2 - a 2 - b 2 On using compendo and dividendo rule for the given, we get ⇒ x + 1 x - 1 = 2 a 2 + b 2 2 a 2 - b 2 Squaring on both the sides, we get ⇒ ( x + 1 ) 2 x - 1 2 = a 2 + b 2 2 a 2 - b 2 2 ⇒ x 2 + 2 x + 1 x 2 - 2 x + 1 = a 2 + b 2 a 2 - b 2 ⇒ x 2 + 2 x + 1 + x 2 - 2 x + 1 x 2 + 2 x + 1 - x 2 - 2 x + 1 = a 2 + b 2 + a 2 - b 2 a 2 + b 2 - a 2 - b 2 ⇒ b 2 = 4 a 2 x 2 x 2 + 1 ⇒ b 2 = 2 a 2 x x 2 + 1 Hence, it is proved.

E M B I B E

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.3>Q 15

MEDIUM

10th ICSE

IMPORTANT

Earn 100

Given $x = \frac{\sqrt{a^{2} + b^{2}} + \sqrt{a^{2} - b^{2}}}{\sqrt{a^{2} + b^{2}} - \sqrt{a^{2} - b^{2}}}$ , use componendo and dividendo to prove that $b^{2} = \frac{2 a^{2} x}{x^{2} + 1}$ .

Important Questions on Ratio and Proportion

EASY

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.3>Q 16

Given that

\frac{a^{3} + 3 a b^{2}}{b^{3} + 3 a^{2} b} = \frac{63}{62}

using Componendo and Dividendo, find

a : b

.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.3>Q 17

Given

\frac{x^{3} + 12 x}{6 x^{2} + 8} = \frac{y^{3} + 27 y}{9 y^{2} + 27}

, using componendo and dividendo, find

x : y

.

EASY

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.3>Q 18

Using the properties of proportion, solve the following equation for $x$ .

$\frac{x^{3} + 3 x}{3 x^{2} + 1} = \frac{341}{91}$

EASY

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.3>Q 19

If $\frac{x + y}{a x + b y} = \frac{y + z}{a y + b z} = \frac{z + x}{a z + b x}$ , Prove that each of these ratios is equal to $\frac{2}{a + b}$ , unless $x + y + z = 0$ .