Question : If the equation k(21x2 + 24) + rx + (14x2 – 9) = 0, k(7x2 + 8) + px + (2x2 – 3) = 0 have both roots common, then the value of $\frac{p}{r}$ is:
Option 1: $\frac{1}{3}$
Option 2: $\frac{2}{5}$
Option 3: $\frac{4}{3}$
Option 4: $\frac{7}{5}$
Correct Answer: $\frac{1}{3}$
Solution :
Since the roots of the equation are common, the coefficients are in proportion.
The equation 1 can be written as, (21k +14)x
2
+ rx + 24k – 9 = 0
The equation 2 can be written as, (7k + 2)x
2
+ px + 8k – 3 = 0
Therefore,
$\frac{21k+14}{7k+2} = \frac{r}{p} = \frac{24k-9}{8k-3}$
$⇒\frac{r}{p} = \frac{24k−9}{8k−3}$
$⇒\frac{r}{p} = \frac{3 \times (8k−3)}{(8k−3)}$
$⇒\frac{r}{p} = \frac{3}{1}$
$\therefore\frac{p}{r}=\frac{1}{3}$
Hence, the correct answer is $\frac{1}{3}$.
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