The equation of the ellipse with focus -1-1- directrix x-y-3-0 and eccentricity 1 - 2 is A- 7 x2-2 x y-7 y2-10 x-10 y-7-0B- 7 x2-2 x y-7 y2-10 x-10 y-7-0C- none of theseD- 7 x2-7 y2-2 x y-10 y-10 x-7-0

You visited us 1 times! Enjoying our articles? Unlock Full Access!

General Equation of Ellipse

The equation ...

Question

The equation of the ellipse with focus (-1,1), directrix x-y+3=0 and eccentricity 1/2 is

7 x^{2} + 2 x y + 7 y^{2} + 10 x + 10 y + 7 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

7 x^{2} + 7 y^{2} + 2 x y - 10 y + 10 x + 7 = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

7 x^{2} + 2 x y + 7 y^{2} + 10 x - 10 y - 7 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

none of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $7 x^{2} + 7 y^{2} + 2 x y - 10 y + 10 x + 7 = 0$

Let P(x,y) be any point on the ellipse whose focus and eccentricity are S(-1,1) and $e = \frac{1}{2}$ , respectively.
Let PM be the perpendicular from P on the directrix.
Then $S P = e \times P M$
$\Rightarrow S P = \frac{1}{2} \times P M$
$\Rightarrow 2 S P = P M$
$\Rightarrow 4 (S P)^{2} = P M^{2}$
$4 [(x + 1)^{2} + (y - 1)^{2}] = {(\frac{x - y + 3}{\sqrt{1^{2} + (- 1)^{2}}})}^{2}$
$\Rightarrow 4 [(x + 1 + 2 x + y^{2} + 1 - 2 y)] = \frac{x^{2} + y^{2} + 9 - 2 x y - 6 y + 6 x}{2}$
$\Rightarrow 8 x^{2} + 8 + 16 x + 8 y^{2} + 8 - 16 y = x^{2} + y^{2} + 9 - 2 x y - 6 y + 6 x$
$\Rightarrow 7 x^{2} + 7 y^{2} + 2 x y - 10 y + 10 x + 7 = 0$
This is the required equation of the ellipse.

Suggest Corrections

Similar questions

Q. The equation of the ellipse with focus (−1, 1), directrix x − y + 3 = 0 and eccentricity 1/2 is
(a) 7x² + 2xy + 7y² + 10x + 10y + 7 = 0
(b) 7x² + 2xy + 7y² + 10x − 10y + 7 = 0
(c) 7x² + 2xy + 7y² + 10x − 10y − 7 = 0
(d) none of these

Q. The equation of an ellipse with focus at

(1, - 1),

directrix

x - y - 3 = 0

and eccentricity

\frac{1}{2}

Q. The equation of the parabola whose focus is (1, −1) and the directrix is x + y + 7 = 0 is
(a) x² + y² − 2xy − 18x − 10y = 0
(b) x² − 18x − 10y − 45 = 0
(c) x² + y² − 18x − 10y − 45 = 0
(d) x² + y² − 2xy − 18x − 10y − 45 = 0

The equation of the parabola whose focus is (1,-1) and the directrix is x+y+7=0 is

Match the column

$\begin{matrix} Equation & Name of the curve 1) x^{2} - 2 x - y - 3 = 0 & P) Circle 2) x^{2} + 3 x y + 2 y^{2} - x - 4 y - 6 = 0 & Q) Parabola 3) x^{2} + y^{2} - 20 = 0 & R) Ellipse 4) 7 x^{2} + 7 y^{2} + 2 x y + 10 x - 10 y + 7 = 0 & S) Hyperbola 5) 6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0 & T) Pair of straight lines \end{matrix}$

Join BYJU'S Learning Program