Question:
The ratio by which the line \( 2x + 5y - 7 = 0 \) divides the straight line joining the points \( (-4, 7) \) and \( (6, -5) \) is:
The ratio by which the line \( 2x + 5y - 7 = 0 \) divides the straight line joining the points \( (-4, 7) \) and \( (6, -5) \) is:
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If the calculated values at two points have opposite signs, the line divides the segment internally.
Updated On: May 6, 2026
- \( 1 : 4 \)
- \( 1 : 2 \)
- \( 1 : 1 \)
- \( 2 : 3 \)
- \( 1 : 3 \)
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The Correct Option is C
Solution and Explanation
Concept:
If a line \[ ax+by+c=0 \] divides the line segment joining \[ (x_1,y_1) \text{ and } (x_2,y_2), \] then the ratio is given by: \[ \text{Ratio} = -\frac{ax_1+by_1+c}{ax_2+by_2+c} \]
Step 1: Identify the given values.
Line: \[ 2x+5y-7=0 \] Points: \[ P_1(-4,7), \qquad P_2(6,-5) \]
Step 2: Calculate value for first point.
\[ L_1 = 2(-4)+5(7)-7 \] \[ =-8+35-7 \] \[ =20 \]
Step 3: Calculate value for second point.
\[ L_2 = 2(6)+5(-5)-7 \] \[ =12-25-7 \] \[ =-20 \]
Step 4: Find the ratio.
\[ \text{Ratio} = -\frac{20}{-20} \] \[ =1 \] Hence, the line divides the segment in the ratio: \[ 1:1 \] Final Answer:
\[ \boxed{1:1} \]
If a line \[ ax+by+c=0 \] divides the line segment joining \[ (x_1,y_1) \text{ and } (x_2,y_2), \] then the ratio is given by: \[ \text{Ratio} = -\frac{ax_1+by_1+c}{ax_2+by_2+c} \]
Step 1: Identify the given values.
Line: \[ 2x+5y-7=0 \] Points: \[ P_1(-4,7), \qquad P_2(6,-5) \]
Step 2: Calculate value for first point.
\[ L_1 = 2(-4)+5(7)-7 \] \[ =-8+35-7 \] \[ =20 \]
Step 3: Calculate value for second point.
\[ L_2 = 2(6)+5(-5)-7 \] \[ =12-25-7 \] \[ =-20 \]
Step 4: Find the ratio.
\[ \text{Ratio} = -\frac{20}{-20} \] \[ =1 \] Hence, the line divides the segment in the ratio: \[ 1:1 \] Final Answer:
\[ \boxed{1:1} \]
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