Exam

JEE Main

Year

2015

Date

10 April

Mode

Online

Subject

Mathematics

Chapter

Circle

JEE Main 2015 · 10 April · Online · Mathematics · Circle

Let the tangents drawn to the circle, x ^ 2 + y ^ 2 = 16 from the...

Let the tangents drawn to the circle, ${x}^{2} + {y}^{2} = 16$ from the point$$P\left(0, h\right)$ meet the $x$-axis at points $A and$B$. If the area of$\delta APB$is minimum, then positive value of$h$ is:

Step-by-step solution

Let equation of tangent be $xcos \alpha + ysin \alpha = 4$

Now, its points of intersection with co-ordinate axis are

$A\left(\dfrac{4}{cos \alpha}, 0\right), P\left(0, \dfrac{4}{sin \alpha}\right)$

Now, $ar\left(∆ PAB\right) = 2\left(ar ∆ PAO\right)$

$= 2 \times \dfrac{1}{2}\left(OA \times OP\right)$

$= \left|\dfrac{4}{sin \alpha} . \dfrac{4}{cos \alpha}\right| = \left|\dfrac{32}{\sin 2\alpha}\right| \ge 32$

which is maximum if, $\alpha = \dfrac{\pi}{4}$

$∴ P\left(0, \dfrac{4}{sin \dfrac{\pi}{4}}\right)$

$∴ h = 4\sqrt{2}$