14. The sum to numbers is 8 and the sum of their reciprocals is 1 Find the numbers.
15. In figure ZBED = ZBDE and E divides BC in ratio 2:1. Prove that AF BE = 2AD CF.
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Answer:
Let the two numbers be x and y
Then
=> x + y = 8 -----------(1)
=> x = 8 - y ----------------(2)
Also, given that sum of reciprocals
= > \frac{1}{x} + \frac{1}{y} = \frac{8}{15}=>
x
1
+
y
1
=
15
8
= > \frac{x + y}{xy} = \frac{8}{15}=>
xy
x+y
=
15
8
Replacing (1) in above equation
= > \frac{8}{xy} = \frac{8}{15}=>
xy
8
=
15
8
Cancelling 8 on Numerator sides and cross Multiplying
=> xy = 15
Using (2) in above equation
=> y(8 - y) = 15
=> y² - 8y + 15 = 0
By middle term splitting method
=> y² - 3y - 5y + 15 = 0
=> y(y - 3) - 5(y - 3) = 0
Thus
=> (y - 3)(y - 5) = 0
Thus y = 3 or y = 5
Using this in (2) we have
Thus x = 5 or x = 3
Thus the two numbers are 3 and 5
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