An ellipse has eccentricity (1/2) and one focus at S ((1/2), 1). Its one directrix is common tangent (nearer to S ) to the circle x2+y2=1 and x2-y2=1. The equation of the ellipse is

Question

An ellipse has eccentricity (1/2) and one focus at S ((1/2), 1). Its one directrix is common tangent (nearer to S ) to the circle x2+y2=1 and x2-y2=1. The equation of the ellipse is (A) 9(x-(1/3))2+12(y-1)2=1 (B) 12(x-(1/3))2+9(y-1)2=1 (C) 3(x-(1/3))2+4(y-1)2=1 (D) 9(x-(1/3))2+16(y-1)2=1

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/b1c312b653117d3e4286b55fb29c411c-.png" /> Directrix ⇒ x =1 text focus =((1/2), 1) e =(1/2) Θ ( PS / PM )= e ⇒ PS 2= e 2 PM 2 ⇒ ( x -(1/2))2+( y -1)2=((1/2))2( x -1)2 ⇒ 9( x -(1/3))2+12( y -1)2=1

Q. An ellipse has eccentricity $\frac{1}{2}$ and one focus at $S (\frac{1}{2}, 1)$ . Its one directrix is common tangent (nearer to $S$ ) to the circle $x^{2} + y^{2} = 1$ and $x^{2} - y^{2} = 1$ . The equation of the ellipse is

$9 (x - \frac{1}{3})^{2} + 12 (y - 1)^{2} = 1$

$12 (x - \frac{1}{3})^{2} + 9 (y - 1)^{2} = 1$

$3 (x - \frac{1}{3})^{2} + 4 (y - 1)^{2} = 1$

$9 (x - \frac{1}{3})^{2} + 16 (y - 1)^{2} = 1$

Directrix $\Rightarrow x = 1$
$focus = (\frac{1}{2}, 1)$
$e = \frac{1}{2}$
$Θ \frac{PS}{PM} = e \Rightarrow P S^{2} = e^{2} P M^{2}$
$\Rightarrow (x - \frac{1}{2})^{2} + (y - 1)^{2} = (\frac{1}{2})^{2} (x - 1)^{2}$
$\Rightarrow 9 (x - \frac{1}{3})^{2} + 12 (y - 1)^{2} = 1$

Q. An ellipse has eccentricity 21​ and one focus at S(21​,1). Its one directrix is common tangent (nearer to S ) to the circle x2+y2=1 and x2−y2=1. The equation of the ellipse is

9(x−31​)2+12(y−1)2=1

12(x−31​)2+9(y−1)2=1

3(x−31​)2+4(y−1)2=1

9(x−31​)2+16(y−1)2=1