Question:
If a point \((\alpha,\beta)\) of \(3x+y=0\) and \((3,4)\) lie on the opposite sides of \(3x-4y-8=0\), then which of the following is correct?
If a point \((\alpha,\beta)\) of \(3x+y=0\) and \((3,4)\) lie on the opposite sides of \(3x-4y-8=0\), then which of the following is correct?
Show Hint
If two points lie on opposite sides of a line
\[
ax+by+c=0,
\]
then the expressions obtained after substituting the points into
\[
ax+by+c
\]
must have opposite signs.
Updated On: Jun 25, 2026
- \(15\alpha-8\gt 0\)
- \(\alpha \in (-\infty,\infty)\)
- \(15\alpha-8=0\)
- \(\alpha=0\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Use the condition that \((\alpha,\beta)\) lies on the line \(3x+y=0\).
Since the point \((\alpha,\beta)\) lies on \[ 3x+y=0, \] we substitute \(x=\alpha\) and \(y=\beta\): \[ 3\alpha+\beta=0 \] Therefore, \[ \beta=-3\alpha \] Hence, the point becomes \[ (\alpha,-3\alpha) \]
Step 2: Use the opposite side condition.
For two points to lie on opposite sides of the line \[ 3x-4y-8=0, \] the values obtained by substituting the points into \[ f(x,y)=3x-4y-8 \] must have opposite signs.
First, substitute the point \((3,4)\): \[ f(3,4)=3(3)-4(4)-8 \] \[ =9-16-8 \] \[ =-15 \] Now substitute the point \((\alpha,-3\alpha)\): \[ f(\alpha,-3\alpha)=3\alpha-4(-3\alpha)-8 \] \[ =3\alpha+12\alpha-8 \] \[ =15\alpha-8 \]
Step 3: Apply the opposite sign condition.
Since the two points are on opposite sides, \[ f(3,4)\cdot f(\alpha,-3\alpha)\lt 0 \] Therefore, \[ (-15)(15\alpha-8)\lt 0 \] Dividing both sides by \(-15\) reverses the inequality: \[ 15\alpha-8\gt 0 \]
Step 4: Final conclusion.
Hence, the correct option is \[ \boxed{15\alpha-8\gt 0} \]
Since the point \((\alpha,\beta)\) lies on \[ 3x+y=0, \] we substitute \(x=\alpha\) and \(y=\beta\): \[ 3\alpha+\beta=0 \] Therefore, \[ \beta=-3\alpha \] Hence, the point becomes \[ (\alpha,-3\alpha) \]
Step 2: Use the opposite side condition.
For two points to lie on opposite sides of the line \[ 3x-4y-8=0, \] the values obtained by substituting the points into \[ f(x,y)=3x-4y-8 \] must have opposite signs.
First, substitute the point \((3,4)\): \[ f(3,4)=3(3)-4(4)-8 \] \[ =9-16-8 \] \[ =-15 \] Now substitute the point \((\alpha,-3\alpha)\): \[ f(\alpha,-3\alpha)=3\alpha-4(-3\alpha)-8 \] \[ =3\alpha+12\alpha-8 \] \[ =15\alpha-8 \]
Step 3: Apply the opposite sign condition.
Since the two points are on opposite sides, \[ f(3,4)\cdot f(\alpha,-3\alpha)\lt 0 \] Therefore, \[ (-15)(15\alpha-8)\lt 0 \] Dividing both sides by \(-15\) reverses the inequality: \[ 15\alpha-8\gt 0 \]
Step 4: Final conclusion.
Hence, the correct option is \[ \boxed{15\alpha-8\gt 0} \]
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