If a + c = m b and 1 b + 1 d = m c , prove that a , b , c , d are in proportion.

Given, a + c = m b and 1 b + 1 d = m c , ⇒ a + c = m b a c + c d = m . . . . ( dividing by b) . . . ( i ) and a c + c d = m c c b + c d = m ( multiplying by c ) . . . ( i i ) from (i) and (ii) = a b + c b = c b + c d ⇒ a b = c d Hence, a , b , c , d are in proportion.

If x a = y b = z c , prove that x 3 a 2 + y 3 b 2 + z 3 c 2 = ( x + y + z ) 3 ( a + b + c ) 2 .

Given, x a = y b = z c We can write it as x = a k , y = b k and z = c k LHS = x 3 a 2 + y 3 b 2 + z 3 c 2 It can be written as = a 3 k 3 a 2 + b 3 k 3 b 2 + c 3 k 3 c 2 So we get = a k 3 + b k 3 + c k 3 Taking common terms = k 3 ( a + b + c ) RHS = ( x + y + z ) 3 ( a + b + c ) 2 It can be written as = ( a k + b k + c k ) 3 ( a + b + c ) 2 So we get = k 3 ( a + b + c ) 3 ( a + b + c ) 2 = k 3 ( a + b + c ) Therefore, it is proved L H S = R H S .

If x a = y b = z c , prove that ( a 2 x 2 + b 2 y 2 + c 2 z 2 a 3 x + b 3 y + c 3 z ) 3 = x y z a b c

Let x a = y b = z c = k . ∴ x = a k , y = b k and z = c k . Now, a 3 + b 3 + c 3 x 3 + y 3 + z 3 = a 3 + b 3 + c 3 a k 3 + b k 3 + c k 3 = a 3 + b 3 + c 3 a 3 k 3 + b 3 k 3 + c 3 k 3 = a 3 + b 3 + c 3 k 3 a 3 + b 3 + c 3 = 1 k 3 And, a b c x y z = a b c a k b k c k = a b c a b c k 3 = 1 k 3 Therefore, a 3 + b 3 + c 3 x 3 + y 3 + z 3 = a b c x y z Hence, proved.

If x a = y b = z c , prove that a x - b y ( a + b ) ( x - y ) + b y - c z ( b + c ) ( y - z ) + c z - a x ( c + a ) ( z - x ) = 3

Given, LHS = a x - b y ( a + b ) ( x - y ) + b y - c z ( b + c ) ( y - z ) + c z - a x ( c + a ) ( z - x ) It can be written as = a x - b y ( a + b ) ( a k - b k ) + b y - c z ( b + c ) ( b k - c k ) + c z - a x ( c + a ) ( c k - a k ) By further calculation = a 2 k - b 2 k ( a + b ) k ( a - b ) + b 2 k - c 2 k ( b + c ) k ( b - c ) + c 2 k - a 2 k ( c + a ) k ( c - a ) Taking common terms = k a 2 - b 2 k a 2 - b 2 + k b 2 - c 2 k b 2 - c 2 + k c 2 - a 2 k c 2 - a 2 So we get = 1 + 1 + 1 = 3 = R . H . S Therefore, hence proved that LHS = RHS

If a b = c d = e f , prove that ( b 2 + d 2 + f 2 ) ( a 2 + c 2 + e 2 ) = ( a b + c d + e f ) 2 .

Let us Consider a b = c d = c f = k So we get, a = b k , c = d k , e = f k LHS = b 2 + d 2 + f 2 a 2 + c 2 + e 2 We can write it as, = b 2 + d 2 + f 3 b 2 k 2 + d 2 k 2 + f 2 k 2 Taking out the common terms = b 2 + d 2 + f 2 k 2 b 2 + d 2 + f 2 then we get = k 2 b 2 + d 2 + f 2 RHS = ( a b + c d + e f ) 2 We can write it as = ( b , k b + d k , d + f k , f ) 2 = k b 2 + k d 2 + k f ' By Taking out common terms = k 2 b 2 + d 2 + f 2 2 Therefore, hence proved that LHS = RHS .

If a b = c d = e f , prove that ( a 3 + c 3 ) 2 ( b 3 + d 3 ) 2 = e 6 f 6 .

Let us consider, a b = c d = e f = k ∴ a = b k , c = d k , e = f k L.H.S a 3 + c 3 2 b 3 + d 3 2 = b 3 k 3 + d 3 k 3 2 b 3 + d 3 2 = k 3 b 3 + d 3 2 b 3 + d 3 2 = k 6 b 3 + d 3 b 3 + d 3 = k 6 R.H.S = e 6 f 6 = f 6 k 6 f 6 = k 6 Therefore, it is proved that, L.H.S = R.H.S

If a b = c d = e f , prove that a 2 b 2 + c 2 d 2 + e 2 f 2 = a c b d + c e d f + a e b f .

Given, a b = c d = e f = k a = b k ; c = d k ; e = f k LHS = a 2 b 2 + c 2 d 2 + e 2 f 2 It can be written as = b 2 k 2 b 2 + d 2 k 2 d 2 + f 2 k 2 f 2 So we get = k 2 + k 2 + k 2 = 3 k 2 RHS = a c b d + c e d f + a e b f It can be written as = b k · d k b d + d k · f k d f + b k · f k b f So we get = k 2 + k 2 + k 2 = 3 k 2 Therefore, it is proved that LHS = RHS .

E M B I B E

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.2>Q 13

HARD

10th ICSE

IMPORTANT

Earn 100

If $b$ is the mean proportional between $a$ and $c,$ prove that $(a b + b c)$ is the mean proportional between $(a^{2} + b^{2}) and (b^{2} + c^{2}) .$

Important Questions on Ratio and Proportion

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.2>Q 14

If $y$ is the mean proportional between $x and z$ , prove that $x y z {(x + y + z)}^{3} = {(x y + y z + z x)}^{3}$ .

MEDIUM

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.2>Q 15

If

a + c = m b and \frac{1}{b} + \frac{1}{d} = \frac{m}{c},

prove that

a, b, c, d

are in proportion.

HARD

10th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 10>Chapter 7 - Ratio and Proportion>Exercise 7.2>Q 16

If $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$ , prove that $\frac{x^{3}}{a^{2}} + \frac{y^{3}}{b^{2}} + \frac{z^{3}}{c^{2}} = \frac{{(x + y + z)}^{3}}{{(a + b + c)}^{2}}$ .