Question

If one zero of the polynomial $p\left( x \right) = \left( {k + 4} \right){x^2} + 13x + 3k$ is the reciprocal of the other, then k is equal to
$
  (a){\text{ 2}} \\
  (b){\text{ 3}} \\
  (c){\text{ 4}} \\
  (d){\text{ 5}} \\
$

Answer 1

Hint: In this question the relation between the roots of the given polynomial which is a quadratic equation is given to us. Assume one root to be variable, use the given condition and use the concept of product of roots formula for a quadratic equation to get the value of k.

Complete step-by-step answer:

Given polynomial $p\left( x \right) = \left( {k + 4} \right){x^2} + 13x + 3k$
Now it is given that the one zero of the polynomial is reciprocal of the other.

And we all know that zero is nothing but the root of the polynomial.
So let the first root of the polynomial be $\alpha $ so the second root according to given information is the reciprocal of the first root therefore the second root is $\dfrac{1}{\alpha }$.

So, the product of roots is $\alpha \times \dfrac{1}{\alpha } = 1$.

Now as we know for the quadratic equation $a{x^2} + bx + c = 0$ the product of roots is the ratio of constant term to the coefficient of x.

Therefore product of roots =$\dfrac{c}{a}$.
So on comparing from the given quadratic equation.
$ \Rightarrow c = 3k,a = \left( {k + 4} \right)$

Therefore the product of roots of the given equation is =$\dfrac{{3k}}{{k + 4}}$.
And the product of roots of the given equation is 1 as we calculated.
$ \Rightarrow 1 = \dfrac{{3k}}{{k + 4}}$

Now simplify this equation we have,
$ \Rightarrow 3k = k + 4$
$ \Rightarrow 2k = 4$

Now divide by 2 throughout we have,
$ \Rightarrow k = 2$
So this is the required value of k.

Hence option (A) is correct.

Note: Whenever we come across such types of problems in which the relation between the roots are given in case of a quadratic equation, always remember that the concept of sum of roots and the product of roots will eventually be applicable to the given problem statement. Use this to get on the right track to reach the answer.