Question

Let alpha , beta be the roots of \[{x^2} - x + p = 0\] and gamma and delta be roots of \[{x^2} - 4x + q = 0\] . If alpha , beta , gamma, delta are in G.P , then integral values of \[p\] and \[q\]are respectively
A.\[ - 2, - 32\]
B.\[ - 2, - 3\]
C.\[ - 6, - 3\]
D.\[ - 6, - 32\]

Answer 1

Hint: Since the roots are in G.P , therefore we have to find the sum and product of the roots for both equations respectively . The sum of the roots of a quadratic equation \[a{x^2} + bx + c = 0\] is given by \[sum = \dfrac{{ - b}}{a}\] and product of the roots is given by \[product = \dfrac{c}{a}\] , where \[a,b,c\] are the coefficients of quadratic equation .

Complete step-by-step answer:
Sum of the roots of the equation \[{x^2} - x + p = 0\] is given by ,
\[\alpha + \beta = 1\] , where \[\alpha \] and \[\beta \] are the roots of the equation .
Now the product of the roots of same equation is given by
\[\alpha \beta = \dfrac{p}{1}\], on simplifying we get ,
\[\alpha \beta = p\] …..equation (a)
Similarly , for the equation \[{x^2} - 4x + q = 0\]
Sum of the roots is given by
\[\gamma + \delta = - \left( {\dfrac{{ - 4}}{1}} \right)\] , on simplifying we get
\[\gamma + \delta = 4\]
Now product of the roots of above equation is given
\[\gamma \delta = \dfrac{q}{1}\]
\[\gamma \delta = q\] ……equation (b)
Now as the roots of both equation are in G.P , therefore the they are represented as terms of G.P as
\[\alpha = a\] , \[\beta = ar\] , \[\gamma = a{r^2}\] ,\[\delta = a{r^3}\] ,
Now , taking equating sum of roots of both equations we have
\[\alpha + \beta = a + ar\] and \[\gamma + \delta = a{r^2} + a{r^3}\] putting the values of sum we get ,
\[a + ar = 1\] ……equation (i)
\[a{r^2} + a{r^3} = 4\] ……..equation (ii) , on solving we get
\[{r^2}\left( {a + ar} \right) = 4\]
Now putting the value from equation (i) we get,
\[{r^2}\left( 1 \right) = 4\]
\[{r^2} = 4\]
Taking square root on both sides we get ,
\[r = \pm 2\]
Now putting \[r = 2\] in equation (i) we get ,
\[a(1 + 2) = 1\]
\[a = \dfrac{1}{3}\]
Now putting the value of \[r = - 2\] in equation (i) we get ,
\[a(1 - 2) = 1\]
\[a = - 1\]
Now for the values of \[p\] and \[q\] we have
When \[r = 2\]
For the value of \[p\]
\[\alpha \beta = p\]
\[p = \left( a \right)\left( {ar} \right)\]
On putting the values we have
\[p = \left( {\dfrac{1}{3}} \right)\left( {\dfrac{1}{3} \times 2} \right)\]
\[p = \dfrac{2}{9}\]
Now for the value of \[q\] we have ,
\[\gamma \delta = q\]
\[q = \left( {a{r^2}} \right)\left( {a{r^3}} \right)\]
On putting the values we have ,
\[q = \left( {\dfrac{1}{3} \times {2^2}} \right)\left( {\dfrac{1}{3} \times {2^3}} \right)\]
On further solving we get
\[q = \left( {\dfrac{4}{3}} \right)\left( {\dfrac{8}{3}} \right)\]
\[q = \dfrac{{32}}{9}\]
Now when \[r = - 2\] we have
For the value of \[p\]
\[\alpha \beta = p\]
\[p = \left( a \right)\left( {ar} \right)\]
On putting the values
\[p = \left( { - 1} \right)\left[ {\left( { - 1} \right)\left( { - 2} \right)} \right]\]
\[p = - 2\]
Now for the value of \[q\] we have ,
\[\gamma \delta = q\]
\[q = \left( {a{r^2}} \right)\left( {a{r^3}} \right)\]
Putting the given values,
\[q = \left( {\left( { - 1} \right){{\left( { - 2} \right)}^2}} \right)\left( {\left( { - 1} \right){{\left( { - 2} \right)}^3}} \right)\]
On simplifying we get
\[q = \left( { - 4} \right)\left( 8 \right)\]
\[q = - 32\]
In the question it is asked for integer values so the required answer will \[p = - 2\] and \[q = - 32\] .
Therefore , option (1) is the correct answer .
So, the correct answer is “Option 1”.

Note: Whenever , the terms are in Geometric Progression you always have to calculate the product of the roots . Geometric progression has a common ratio between successive terms denoted by \[r\].