SATHEE: Ellipse 2 Question 6

Ellipse 2 Question 6

6. If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ at all points on the ellipse other than its four vertices, then the mid-points of the tangents intercepted between the coordinate axes lie on the curve

(2019 Main, 11 Jan I)

(a) $\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1$

(b) $\frac{1}{4 x^{2}} + \frac{1}{2 y^{2}} = 1$

(d) $\frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$

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Solution:

Given equation of ellipse is $x^{2} + 2 y^{2} = 2$ , which can be written as $\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1$

Let $P$ be a point on the ellipse, other than its four vertices. Then, the parametric coordinates of $P$ be $(\sqrt{2} \cos θ, \sin θ)$

Now, the equation of tangent at $P$ is $\frac{x \sqrt{2} \cos θ}{2} + \frac{y \sin θ}{1} = 1$

$[∵$ equation of tangent at $(x_{1}, y_{1})$ is given by $T = 0$

$\Rightarrow \frac{x x_{1}}{a^{2}} + \frac{y y_{1}}{b^{2}} = 1$

$\Rightarrow \frac{x}{\sqrt{2} \sec θ} + \frac{y}{cosec θ} = 1$

$∴ A (\sqrt{2} \sec θ, 0)$ and $B (0, cosec θ)$

Let mid-point of $A B$ be $R (h, k)$ , then

$\begin{aligned} h & = \frac{\sqrt{2} \sec θ}{2} and k = \frac{cosec θ}{2} \\ 2 h & = \sqrt{2} \sec θ and 2 k = cosec θ \\ \Rightarrow \cos θ & = \frac{1}{\sqrt{2} h} and \sin θ = \frac{1}{2 k} \end{aligned}$

We know that, $\cos^{2} θ + \sin^{2} θ = 1$

$∴ \frac{1}{2 h^{2}} + \frac{1}{4 k^{2}} = 1$

So, locus of $(h, k)$ is $\frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$

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Ellipse 2 Question 6

6. If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ at all points on the ellipse other than its four vertices, then the mid-points of the tangents intercepted between the coordinate axes lie on the curve

Solution:

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Ellipse 2 Question 6

6. If tangents are drawn to the ellipse x2+2y2=2 at all points on the ellipse other than its four vertices, then the mid-points of the tangents intercepted between the coordinate axes lie on the curve

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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6. If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ at all points on the ellipse other than its four vertices, then the mid-points of the tangents intercepted between the coordinate axes lie on the curve