Advertisements
Advertisements
प्रश्न
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
Advertisements
उत्तर
a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk. k = dk2,
a = bk = dk2. k = dk3
L.H.S. = (a + d)(b + c) – (a + c)(b + d)
= (dk3 + d) (dk2 + dk) – (dk3 + dk) (dk2 + d)
= d(k3 + 1) dk(k + 1) – dk (k2 + 1) d(k2 + 1)
= d2k(k + 1) (k3 + 1) – d2k (k2 + 1) (k2 + 1)
= d2k[k4 + k3 + k + 1 – k4 - 2k2 - 1]
= d2k[k3 – 2k2 + k]
= d2k2[k2 – 2k + 1]
= d2k2(k – 1)2
R.H.S. = (b – c)2
= (dk2 – dk)2
= d2k2(k – 1)2
∴ L.H.S. = R.H.S.
Hence proved.
संबंधित प्रश्न
6 is the mean proportion between two numbers x and y and 48 is the third proportional of x and y. Find the numbers.
If (a2 + b2)(x2 + y2) = (ax + by)2; prove that: `a/x = b/y`.
Find the value of x in the following proportions : 3 : x = 24 : 2
Find the mean proportion of: 5 and 80
The sides of a triangle are in the ratio 1 : 3 : 5 and its perimeter is 90 cm. The length of its largest side is
If a, b, c and d are in proportion, prove that: `(a + c)^3/(b + d)^3 = (a(a - c)^2)/(b(b - d)^2)`
If a, b, c are in continued proportion, prove that: a : c = (a2 + b2) : (b2 + c2)
A student said that the ratios `3/4` and `9/16` were proportional. What error did the student make?
Which of the following statements correctly defines a proportion involving four non-zero quantities a, b, c, and d?
A picture frame’s length and width are in the ratio 4 : 3. If the length is 28 cm, what is the width?
