Question

If the pair of lines $3x^{2}-5xy+py^{2}=0$ and $6x^{2}-xy-5y^{2}=0$ have one line common, then $p=$

• A. $2, \frac{25}{4}$
• B. $-2, 2$
• C. $2, \frac{-25}{4}$
• D. $\frac{-25}{4}$

Hint:

Substitute the slope $m$ from the known lines into the auxiliary equation $bm^{2}+2hm+a=0$.

Answer 1

The Correct Option is C

Step 1: Concept
Factorize the known equation to find the individual lines.

Step 2: Analysis
$6x^{2}-xy-5y^{2}=0 \implies (6x+5y)(x-y)=0$.
The common line is either $x-y=0$ or $6x+5y=0$.

Step 3: Calculation
Case 1: $y=x$. Substitute in $3x^{2}-5x(x)+px^{2}=0 \implies 3-5+p=0 \implies p=2$.
Case 2: $y=-6x/5$. Substitute in $3x^{2}-5x(-6x/5)+p(-6x/5)^{2}=0 \implies 3+6+p(36/25)=0$
$9 + 36p/25 = 0 \implies p = -25/4$.

Step 4: Conclusion
Hence, $p = 2, -25/4$. Final Answer: (C)