SATHEE: Ellipse 2 Question 10

Ellipse 2 Question 10

10. The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2} + 3 y^{2} = 6$ on any tangent to it is

(a) ${(x^{2} - y^{2})}^{2} = 6 x^{2} + 2 y^{2}$

(2014 Main)

(b) ${(x^{2} - y^{2})}^{2} = 6 x^{2} - 2 y^{2}$

(d) ${(x^{2} + y^{2})}^{2} = 6 x^{2} - 2 y^{2}$

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Solution:

Equation of ellipse is $x^{2} + 3 y^{2} = 6$ or $\frac{x^{2}}{6} + \frac{y^{2}}{2} = 1$ .

Equation of the tangent is $\frac{x \cos θ}{a} + \frac{y \sin θ}{b} = 1$

Let $(h, k)$ be any point on the locus.

$∴ \frac{h}{a} \cos θ + \frac{k}{b} \sin θ = 1$

Slope of the tangent line is $\frac{- b}{a} \cot θ$ .

Slope of perpendicular drawn from centre $(0, 0)$ to $(h, k)$ is $k / h$ .

Since, both the lines are perpendicular.

$\begin{aligned} ∴ & \frac{k}{h} \times - \frac{b}{a} \cot θ & = - 1 \\ \Rightarrow & \frac{\cos θ}{h a} & = \frac{\sin θ}{k b} = α \\ \Rightarrow & \cos θ & = α h a \\ \sin θ & = α k b \end{aligned}$

From Eq. (i), $\frac{h}{a} (α h a) + \frac{k}{b} (α k b) = 1$

$\begin{aligned} \Rightarrow & h^{2} α + k^{2} α & = 1 \\ \Rightarrow & α & = \frac{1}{h^{2} + k^{2}} \end{aligned}$

Also, $\sin^{2} θ + \cos^{2} θ = 1$

$\Rightarrow (α k b)^{2} + (α h a)^{2} = 1$

$\Rightarrow α^{2} k^{2} b^{2} + α^{2} h^{2} a^{2} = 1$

$\Rightarrow \frac{k^{2} b^{2}}{{(h^{2} + k^{2})}^{2}} + \frac{h^{2} a^{2}}{{(h^{2} + k^{2})}^{2}} = 1$

$\Rightarrow \frac{2 k^{2}}{{(h^{2} + k^{2})}^{2}} + \frac{6 h^{2}}{{(h^{2} + k^{2})}^{2}} = 1 [∵ a^{2} = 6, b^{2} = 2]$

$\Rightarrow 6 x^{2} + 2 y^{2} = {(x^{2} + y^{2})}^{2}$

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अगला

Ellipse 2 Question 10

10. The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2} + 3 y^{2} = 6$ on any tangent to it is

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

Ellipse 2 Question 10

10. The locus of the foot of perpendicular drawn from the centre of the ellipse x2+3y2=6 on any tangent to it is

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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10. The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2} + 3 y^{2} = 6$ on any tangent to it is