The line segment joining (5,0) and (10 cos t, 10 sin t) is divided internally

Question

The line segment joining (5,0) and (10 cos t, 10 sin t) is divided internally in the ratio 2:3 at P. If t varies, then the locus of P is

Options:

A . a straight line

B . a circle

C . an ellipse

D . a hyperbola

Answer: Option B
:
B

P(x,y) divides the line segment joining (5,0) and (10 cos t, 10 sin t) internally in the ratio 2:3
$∴ x = \frac{2 \times 10 c o s t + 3 \times 5}{2 + 3} x = 4 c o s t + 3 y = \frac{2 \times 10 s i n t + 3 \times 0}{2 + 3} y = 4 s i n t$

$(x - 3)^{2} + y^{2} = 16 c o s^{2} t + 16 s i n^{2} t \Rightarrow (x - 3)^{2} + y^{2} = 16$

The above equation represents a circle

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