The equation of the ellipse whose equation of directrix is $3 x + 4 y - 5 = 0$ , coordinates of the focus are $(1, 2)$ and the eccentricity is $\frac{1}{2}$ is $91 x^{2} + 84 y^{2} - 24 x y - 170 x - 360 y + 475 = 0$

True

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

False

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

Open in App

Solution

The correct option is A True
Let $P (x, y)$ be any point on the ellipse and PM be the perpendicular from P upon the directrix $3 x + 4 y - 5 = 0$ .
Then by the definition,
$\frac{S P}{P M} = e$

$S P = e . P M$
$\sqrt{(x - 1)^{2} + (y - 2)^{2}} = \frac{1}{2} | \frac{3 x + 4 y - 5}{\sqrt{3^{2} + 4^{2}}} |$

$(x - 1)^{2} + (y - 2)^{2} = \frac{1}{4} . \frac{(3 x + 4 y - 5)^{2}}{25}$

$100 (x^{2} + y^{2} - 2 x - 4 y + 5) = 9 x^{2} + 16 y^{2} + 24 x y - 30 x - 40 y + 25$
$91 x^{2} + 84 y^{2} - 24 x y - 170 x - 360 y + 475 = 0$ is the equation of the ellipse.

Suggest Corrections

Similar questions

3 x + 4 y - 5 = 0

(1, 2)

\frac{1}{2}

(1, - 1)

\frac{2}{3}

x + y + 2 = 0

(1, 0)

x + y + 1 = 0

\frac{1}{\sqrt{2}}

3 x + 4 y - 5 = 0

x^{2} + y^{2} = (3 x + 4 y - 5)^{2}

Join BYJU'S Learning Program