Given p and q are the roots of the quadratic equation [tex]ax^2-5x+c=0[/tex] with [tex]a\neq0[/tex]. If [tex]p,q,\frac1{8pq}[/tex] forms a geometric sequence and [tex]log_a18+log_ap=1[/tex], the value of c – a is ....
A. [tex]\frac13[/tex]
B. [tex]\frac12[/tex]
C. 3
D. 5
E. 7
Since [tex]p,q,\frac1{8pq}[/tex] is a geometric sequence, then:
[tex]\frac{q}{p}=\frac{\frac1{8pq}}q[/tex]
[tex]\frac{q}{p}=\frac1{8pq^2}[/tex]
[tex]q=\frac1{8q^2}[/tex]
[tex]q^3=\frac18[/tex]
[tex]q=\frac12[/tex]
Also, since [tex]log_a18+log_ap=1[/tex], then:
[tex]log_a18p=log_aa[/tex]
18p = a
[tex]p=\frac{a}{18}[/tex]
This is where the real problem starts. No matter how I substitute, either it will cancel out the a's or p's, or becoming a quadratic equation with no real roots. What should I do?