Question:
If the sum of the slopes of the lines given by \(x^2-2cxy-7y^2=0\) is four times their product, find the value of \(c\).
If the sum of the slopes of the lines given by \(x^2-2cxy-7y^2=0\) is four times their product, find the value of \(c\).
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For homogeneous pair of lines, memorize:
\[
m_1+m_2=-\frac{2h}{b}, \qquad m_1m_2=\frac{a}{b}.
\]
Updated On: Jun 9, 2026
- \(2\)
- \(-2\)
- \(1\)
- \(-1\)
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The Correct Option is A
Solution and Explanation
Concept:
For
\[
ax^2+2hxy+by^2=0
\]
\[
m_1+m_2=-\frac{2h}{b}
\]
and
\[
m_1m_2=\frac{a}{b}
\]
Step 1: Identify coefficients. \[ a=1,\qquad h=-c,\qquad b=-7 \]
Step 2: Find sum and product. \[ m_1+m_2 = -\frac{2(-c)}{-7} = -\frac{2c}{7} \] \[ m_1m_2 = -\frac17 \]
Step 3: Apply the given condition. \[ m_1+m_2 = 4m_1m_2 \] \[ -\frac{2c}{7} = -\frac47 \] \[ c=2 \] Hence \[ \boxed{2} \]
Step 1: Identify coefficients. \[ a=1,\qquad h=-c,\qquad b=-7 \]
Step 2: Find sum and product. \[ m_1+m_2 = -\frac{2(-c)}{-7} = -\frac{2c}{7} \] \[ m_1m_2 = -\frac17 \]
Step 3: Apply the given condition. \[ m_1+m_2 = 4m_1m_2 \] \[ -\frac{2c}{7} = -\frac47 \] \[ c=2 \] Hence \[ \boxed{2} \]
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