Exam

NTA Abhyas

Year

2020

Subject

Mathematics

Chapter

Ellipse

NTA Abhyas 2020 · Mathematics · Ellipse

Let from a point A (h, k) chord of contacts are drawn to the...

Let from a point $A$\left(h, k\right)$chord of contacts are drawn to the ellipse$x^{2} + 2y^{2} = 6$such that all these chords touch the ellipse$x^{2} + 4y^{2} = 4$, then locus of the point$A is

Step-by-step solution

Let chord of contact is $PQ$ that touches ${x}^{2} + 4{y}^{2} = 4$ at $R$

Now, assume $R = \left(2cos \theta, sin \theta\right)$
Equation of $PQ$ is $hx + 2yk = 6$ … \left(1\right)$<br />\left(T = 0forpointA\left(h, k\right)\right)$
Again, the equation of $PQ$ is $\dfrac{xcos\theta}{2} + \dfrac{ysin\theta}{1} = 1$ … \left(2\right)$<br />From \left(1\right)$ & $\left(2\right)$ we get $\dfrac{2h}{cos \theta} = \dfrac{2k}{sin \theta} = 6$
$cos \theta = \dfrac{h}{3}, sin \theta = \dfrac{k}{3} ⇒ {x}^{2} + {y}^{2} = 9$