Question:
A line cuts the positive axes \(OX\), \(OY\) in the points \(A\), \(B\) respectively. If the point \(P(3,5)\) divides the line \(AB\) in the ratio \(2:1\) internally, then the equation of the line is
A line cuts the positive axes \(OX\), \(OY\) in the points \(A\), \(B\) respectively. If the point \(P(3,5)\) divides the line \(AB\) in the ratio \(2:1\) internally, then the equation of the line is
Show Hint
If a line cuts the axes at \((a,0)\) and \((0,b)\), its equation is
\[
\frac{x}{a}+\frac{y}{b}=1.
\]
Use the section formula to determine the intercepts first.
Updated On: Jul 15, 2026
- \(5x+6y=45\)
- \(3x+5y=34\)
- \(5x-3y=0\)
- \(5x-6y+15=0\)
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The Correct Option is A
Solution and Explanation
Step 1: Let the intercepts be \(A(a,0)\) and \(B(0,b)\).
Since \(P(3,5)\) divides \(AB\) internally in the ratio \(2:1\),
\[
P=\left(\frac{2\cdot0+1\cdot a}{3},
\frac{2\cdot b+1\cdot0}{3}\right)
=\left(\frac{a}{3},\frac{2b}{3}\right).
\]
Step 2: Find the intercepts. Comparing coordinates, \[ \frac{a}{3}=3 \quad\Rightarrow\quad a=9, \] \[ \frac{2b}{3}=5 \quad\Rightarrow\quad b=\frac{15}{2}. \]
Step 3: Use the intercept form. \[ \frac{x}{9}+\frac{y}{15/2}=1 \] Multiplying throughout by \(45\), \[ 5x+6y=45. \]
Step 4: Final conclusion. \[ \boxed{5x+6y=45} \] Hence, the correct option is \(\boxed{(A)}\).
Step 2: Find the intercepts. Comparing coordinates, \[ \frac{a}{3}=3 \quad\Rightarrow\quad a=9, \] \[ \frac{2b}{3}=5 \quad\Rightarrow\quad b=\frac{15}{2}. \]
Step 3: Use the intercept form. \[ \frac{x}{9}+\frac{y}{15/2}=1 \] Multiplying throughout by \(45\), \[ 5x+6y=45. \]
Step 4: Final conclusion. \[ \boxed{5x+6y=45} \] Hence, the correct option is \(\boxed{(A)}\).
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