Name: Tangent to a Hyperbola
Uploaded: 2019-07-19
Duration: 2 min 11 s
Description: The tangent to a plane curve at a given point is the straight line that "just touches" the curve at that point. Learn about the tangent to a hyperbola by wat...

Question

Find the lengths of the principal axes, coordinates of the foci, equations of directrices, length of the latus rectum, the distance between foci, the distance between directrices of the ellipse:

$2 x^{2} + 6 y^{2} = 6$

Accepted Answer

Given equation of ellipse is $2 x^{2} + 6 y^{2} = 6$

Dividing both sides by $6$ , we get

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

Comparing this equation with the standard equation of ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$\Rightarrow a^{2} = 3$ and $b^{2} = 1$

$\Rightarrow a = \sqrt{3}$ and $b = 1$

Since $a > b$

$∴$ $X$ -axis is the major axis and the $Y$ -axis is the minor axis i.e., ellipse is of the form as shown in the figure below

Here, the coordinates of $A$ and $B$ are $(- a, 0)$ and $(a, 0)$

Length of major axis $= A B = 2 a = 2 \sqrt{3}$

Length of minor axis $2 x^{2} + 6 y^{2} = 6$

Hence, the principal axes are $2$ and $2 \sqrt{3}$ .

We know that the coordinates of focus of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ are $(a e, 0)$ and $(- a e, 0)$ .

Since for the given ellipse $a = \sqrt{3}$ , we need to find $e$ ,

$e = \frac{c}{a} = \frac{\sqrt{a^{2} - b^{2}}}{a}$ $\Rightarrow e = \frac{\sqrt{3 - 1}}{\sqrt{3}} = \frac{\sqrt{2}}{\sqrt{3}}$ .

Thus, the coordinates of focus are $(\sqrt{3} \times \frac{\sqrt{2}}{\sqrt{3}}, 0)$ and $(- \sqrt{3} \times \frac{\sqrt{2}}{\sqrt{3}}, 0)$ i.e., $(\sqrt{2}, 0)$ and $(- \sqrt{2}, 0)$ .

Equation of directrices are $x = \pm \frac{a}{e}$ i.e. $x = \pm \frac{\sqrt{3}}{\frac{\sqrt{2}}{\sqrt{3}}} = \pm \frac{3}{\sqrt{2}}$ .

Length of latus rectum $= \frac{2 b^{2}}{a} = \frac{2 \times 1}{\sqrt{3}} = \frac{2}{\sqrt{3}}$

Distance between foci $= 2 a e = 2 \times \sqrt{3} \times \frac{\sqrt{2}}{\sqrt{3}} = 2 \sqrt{2}$

Distance between the directrices $= \frac{2 a}{e} = \frac{2 \times \sqrt{3}}{\frac{\sqrt{2}}{\sqrt{3}}} = 3 \sqrt{2}$ .

Find the lengths of the principal axes, coordinates of the foci, equations of directrices, length of the latus rectum, the distance between foci, the distance between directrices of the ellipse:

$2 x^{2} + 6 y^{2} = 6$

Important Questions on Conic Sections

Learn and Prepare for any exam you want !

Find the lengths of the principal axes, coordinates of the foci, equations of directrices, length of the latus rectum, the distance between foci, the distance between directrices of the ellipse: 2x2+6y2=6

Important Questions on Conic Sections

Learn and Prepare for any exam you want !

Find the lengths of the principal axes, coordinates of the foci, equations of directrices, length of the latus rectum, the distance between foci, the distance between directrices of the ellipse:

$2 x^{2} + 6 y^{2} = 6$