Answer:
[tex]13[/tex]
Step-by-step explanation:
Given :-
- The sum of two numbers is 52.
- The product of the numbers is 4 .
And we need to find the sum of reciprocals of the numbers. For that let us assume that the numbers are [tex] x \ \& \ y [/tex] .
According to 1st condition :-
[tex]\implies x + y = 52 [/tex]
Now expressing this equation by keeping only one variable on one side as ,
[tex]\implies x = 52 - y [/tex]
According to 2nd condition :-
[tex]\implies x y = 4 \\\\\implies (52-y)y = 4 \\\\\implies 52 y - y^2 = 4 \\\\\implies y^2 - 52y + 4 = 0 [/tex]
- Now solve this quadratic equation.
On solving the equation using quadratic Formula we will get the value of y as ,
[tex]\implies y = 26\pm 4\sqrt{42}[/tex]
- Note that the sum of x and y is 52 .So in order to find the value of x we will subtract 26±4√42 from 52 and we will get the value of x same as y .
[tex]\implies x = 52 - 26\pm 4\sqrt{42} = 26\pm 4\sqrt{42}[/tex]
Now we need to find out the sum of reciprocal of the two numbers . That will be ,
[tex]\implies \dfrac{1}{26+4\sqrt{42}}+ \dfrac{1}{26-4\sqrt{42}} \\\\\implies \dfrac{26+4\sqrt{42}+26-4\sqrt{42}}{(26+4\sqrt{42})(26-4\sqrt{42}}\\\\\implies \dfrac{52}{(26)^2-(4\sqrt{42})^2} \\\\\implies \dfrac{52}{676-672} \\\\\implies \dfrac{52}{ 4} \\\\\implies \boxed{\red{13}}[/tex]
Hence the sum of their reciprocals is 13 .
