The locus of the points of intersection of the tangents to the circle x=r cos θ, y=r sin θ at points whose parametric angles differ by π / 3, is

Question

The locus of the points of intersection of the tangents to the circle x=r cos θ, y=r sin θ at points whose parametric angles differ by π / 3, is (A) x2+y2=(2/3) r2 (B) x2+y2=(1/3) r2 (C) x2+y2=(4/3) r2 (D) None of these

Tardigrade Admin · Accepted Answer

All such points P satisfying the given condition will be equidistant from the origin

O

(see fig.)

Hence the locus of

P

will be a circle centred at the origin, having radius equal to

OP = \frac{r}{c o s ( \frac{π}{6} )} = \frac{2 r}{3}

Therefore, equation of the
requieed locus is

x^{2} + y^{2} = \frac{4}{3} r^{2}

Q. The locus of the points of intersection of the tangents to the circle $x = r cos θ, y = r sin θ$ at points whose parametric angles differ by $π /3$ , is

$x^{2} + y^{2} = \frac{2}{3} r^{2}$

$x^{2} + y^{2} = \frac{1}{3} r^{2}$

$x^{2} + y^{2} = \frac{4}{3} r^{2}$

None of these

Q. The locus of the points of intersection of the tangents to the circle x=rcosθ,y=rsinθ at points whose parametric angles differ by π/3, is

x2+y2=32​r2

x2+y2=31​r2

x2+y2=34​r2

None of these

All such points P satisfying the given condition will be equidistant from the origin O (see fig.) Hence the locus of P will be a circle centred at the origin, having radius equal to OP=cos(6π​)r​=3​2r​ Therefore, equation of the requieed locus is x2+y2=34​r2

Q. The locus of the points of intersection of the tangents to the circle $x = r cos θ, y = r sin θ$ at points whose parametric angles differ by $π /3$ , is

$x^{2} + y^{2} = \frac{2}{3} r^{2}$

$x^{2} + y^{2} = \frac{1}{3} r^{2}$

$x^{2} + y^{2} = \frac{4}{3} r^{2}$