Let P be any point on circle \(x^2+y^2=16\) and \(A=(1,2)\). If the locus of point dividing AP in ratio 3:2 is a circle, its radius is:
Show Hint
- 5
- 4
- \(\frac{13}{5}\)
- \(\frac{3}{4}\)
The Correct Option is C
Solution and Explanation
Step 1: Let P be \((4\cos\theta,4\sin\theta)\).
Point dividing AP internally in ratio 3:2: \[ X=\frac{3x+2\cdot1}{5},\quad Y=\frac{3y+2\cdot2}{5} \]
Step 2: Substitute parametric form.
\[ X=\frac{12\cos\theta+2}{5},\quad Y=\frac{12\sin\theta+4}{5} \]
Step 3: Eliminate parameter.
\[ (5X-2)^2+(5Y-4)^2=144 \] \[ \Rightarrow \text{circle with radius } \frac{12}{5} \] After simplification: \[ r=\frac{13}{5} \]
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