The minimum area of a triangle formed by any | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the point of tangency be $(h, k)$ on the ellipse $\frac{x^2}{16} + \frac{y^2}{81} = 1$.The equation of the tangent at $(h, k)$ is given by $\f

Vedclass · Accepted Answer

$36$

Vedclass · Answer

(D) Let the point of tangency be $(h, k)$ on the ellipse $\frac{x^2}{16} + \frac{y^2}{81} = 1$.The equation of the tangent at $(h, k)$ is given by $\frac{xh}{16} + \frac{yk}{81} = 1$.The $x$-intercept is found by setting $y=0$,which gives $x = \frac{16}{h}$. So,point $B = (\frac{16}{h}, 0)$.The $y$-intercept is found by setting $x=0$,which gives $y = \frac{81}{k}$. So,point $A = (0, \frac{81}{k})$.The area of the triangle $OAB$ is $A = \frac{1}{2} 	imes |\frac{16}{h}| 	imes |\frac{81}{k}| = \frac{648}{|hk|}$.Since $(h, k)$ lies on the ellipse,$\frac{h^2}{16} + \frac{k^2}{81} = 1$. By the $AM$-$GM$ inequality,$\frac{\frac{h^2}{16} + \frac{k^2}{81}}{2} \ge \sqrt{\frac{h^2 k^2}{16 	imes 81}}$.$\frac{1}{2} \ge \frac{|hk|}{4 	imes 9} \Rightarrow |hk| \le 18$.Therefore,the minimum area $A = \frac{648}{|hk|} \ge \frac{648}{18} = 36$.The minimum area of the triangle is $36$ square units.

The minimum area of a triangle formed by any tangent to the ellipse $\frac{x^2}{16} + \frac{y^2}{81} = 1$ and the coordinate axes is

Similar Questions

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $4x^{2} + 9y^{2} = 36$.

Let the line $y-x=1$ intersect the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{1}=1$ at the points $A$ and $B$. Then the angle subtended by the line segment $AB$ at the center of the ellipse is:

Let $E$ be an ellipse whose major axis is the $X$-axis and minor axis is the $Y$-axis. If the distances of a point $P \left(\frac{5}{2}, 2 \sqrt{3}\right)$ on $E$ from its foci are $\frac{7}{2}$ and $\frac{13}{2}$,then the eccentricity of the ellipse $E$ is

Find the equation of the ellipse $(a > b)$ whose distance between the foci is $8$ and the distance between the directrices is $18$.

If tangents are drawn from any point on the circle $x^2+y^2=25$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$,then the angle between the tangents is

Vedclass Test Series

Exam Paper Generator

Online Exam Module