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# If X/A = Y/B = Z/C Prove that (Ax - By)/((A + B)(X - Y)) + (By - Cz)/((B + C)(Y - Z)) + (Cz - Az)/((C + A)(Z - X)) = 3 - ICSE Class 10 - Mathematics

#### Question

if x/a = y/b = z/c prove that (ax - by)/((a + b)(x - y)) + (by - cz)/((b + c)(y - z)) + (cz - az)/((c + a)(z - x)) = 3

#### Solution

Let x/a = y/b = z/c = k (say)

=> x = ak, y = bk, z = dk

L.H.S

=(ax - by)/((a + b)(x - y)) + (by - cz)/((b + c)(y - z)) + (cz - ax)/((c + a)(z - x))

= (a(ak) - b(bk))/((a + b)(ak - bk)) + (b(bk) - c(ck))/((b + c)(bk - dk)) + (c(ck) - a(ak))/((c + a)(dk - ak))

= (k(a^2 - b^2))/(k(a + b)(a- b)) + (k(b^2 - c^2))/(k(b + c)(b - c)) + (k(c^2 - a^2))/(k(c + a)(c - a))

= (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))

= 1 + 1 + 1 = 3 = R.H.S

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#### APPEARS IN

Selina Solution for Selina ICSE Concise Mathematics for Class 10 (2018-2019) (2017 to Current)
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Ex.7D | Q: 17

#### Video TutorialsVIEW ALL [1]

Solution If X/A = Y/B = Z/C Prove that (Ax - By)/((A + B)(X - Y)) + (By - Cz)/((B + C)(Y - Z)) + (Cz - Az)/((C + A)(Z - X)) = 3 Concept: Ratios.
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