Name: Latus Rectum of an Ellipse
Uploaded: 2019-07-19
Duration: 2 min 46 s
Description: Latus rectum is a special chord of a conic. An Ellipse has two Latus recta. To learn how to draw the latus rectum and know its length, watch the video!

Question

A tangent having slope

- \frac{4}{3}

to the ellipse

\frac{x^{2}}{18} + \frac{y^{2}}{32} = 1

, intersects the axes of

x

and

y

in points

A

and

B

respectively. If

O

is the origin, find the area of triangle

O A B

.

Accepted Answer

We are given,

Equation of ellipse $\frac{x^{2}}{18} + \frac{y^{2}}{32} = 1$

Slope of tangent $x$

We know that the standard equation of ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

On comparing the given equation with the standard equation, we get:

$a^{2} = 18$ and $b^{2} = 32$

We can say that a line $y = m x + c$ is tangent to ellipse only and only if $c^{2} = a^{2} m^{2} + b^{2}$ .

Now on substituting the values in the above equation we have:

$c^{2} = 18 \times \frac{16}{9} + 32$

$\Rightarrow c^{2} = 2 \times 16 + 32$

$\Rightarrow c^{2} = 32 + 32$

$\Rightarrow c^{2} = 64$

$\Rightarrow c = \sqrt{64}$

$\Rightarrow c = 8 . . . (2)$

Now using $(1)$ and $(2)$ we get equation of tangent as,

$y = - \frac{4}{3} x + 8$

From the obtained equation of tangent we get the values of $x$ and $y$ intercepts as $(6, 0) & (0, 8)$ .

So, the height of the triangle is $8$ units and the base is $6$ units. The major axis is $8$ units and the minor axis is $6$ units.

Area of triangle $O A B$ will be

$Area = \frac{1}{2} \times base \times height$

$\Rightarrow Area = \frac{1}{2} \times 6 \times 8$

$\Rightarrow Area = 24 sq . units$

Therefore, the area of triangle $O A B$ is $24$ sq units.

A tangent having slope $- \frac{4}{3}$ to the ellipse $\frac{x^{2}}{18} + \frac{y^{2}}{32} = 1$ , intersects the axes of $x$ and $y$ in points $A$ and $B$ respectively. If $O$ is the origin, find the area of triangle $O A B$ .

Important Questions on Ellipse

Learn and Prepare for any exam you want !

A tangent having slope -43 to the ellipse x218+y232=1, intersects the axes of x and y in points A and B respectively. If O is the origin, find the area of triangle OAB.

Important Questions on Ellipse

Learn and Prepare for any exam you want !

A tangent having slope $- \frac{4}{3}$ to the ellipse $\frac{x^{2}}{18} + \frac{y^{2}}{32} = 1$ , intersects the axes of $x$ and $y$ in points $A$ and $B$ respectively. If $O$ is the origin, find the area of triangle $O A B$ .