If a straight line passing through the point \((2,3)\) intersects X-axis at A and Y-axis at B, then the locus of a point dividing AB in the ratio \(2:3\) is
Show Hint
- \(x^2-5xy+6y^2=0\)
- \(x^2+y^2-4x-6y+4=0\)
- \(x+y=5\)
- \(6x-5xy+6y=0\)
The Correct Option is C
Solution and Explanation
Step 1: Coordinates of dividing point.
Intercepts \[ A(a,0),\qquad B(0,b) \] Point dividing in ratio \(2:3\) \[ P= \left( \frac{2(0)+3a}{5}, \frac{2b+0}{5} \right) \] \[ P= \left( \frac{3a}{5}, \frac{2b}{5} \right) \] Let \[ P(x,y) \] Thus \[ a=\frac{5x}{3} \] \[ b=\frac{5y}{2} \]
Step 2: Substitute relation.
\[ \frac2a+\frac3b=1 \] \[ \frac{2}{5x/3}+\frac{3}{5y/2}=1 \] \[ \frac6{5x}+\frac6{5y}=1 \] After simplification \[ x+y=5 \] Hence \[ \boxed{x+y=5} \]
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