Question

The area of the quadrilateral formed by the tangents at the end points of latus- rectum to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1$, is

• A. 27/4 sq. units
• B. 9 sq. units
• C. 27/2 sq. units
• D. 27sq. units

Answer 1

Answer: (D)

27sq. units

Answer 2

By symmetry the quadrilateral is a rhombus. So area is four times the area of the right angled triangle formed by the tangents and axes in the 1^st quadrant.

Now $a e = \sqrt { a ^ { 2 } - b ^ { 2 } } \Rightarrow a e = 2 \Rightarrow$Tang.

Now $ae = \sqrt{a^{2} - b^{2}} \Rightarrow ae = 2 \Rightarrow$Tangent (in the first quadrant) at one end of latus rectum $\left( 2,\frac{5}{3} \right)$ is

$\frac{2}{9}x + \frac{5}{3}.\frac{y}{5} = 1$

i.e. $\frac{x}{9/2} + \frac{y}{3} = 1$. Therefore area $= 4.\frac{1}{2}.\frac{9}{2}.3 = 27sq.$ units