SATHEE: Application of Derivatives 4 Question 60

Application of Derivatives 4 Question 60

63. A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$ at the points $P$ and $Q$ . If the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ .

If $Δ (h) =$ area of the $△ P Q R, Δ_{1} = max_{1 / 2 \leq h \leq 1} Δ (h)$ and $Δ_{2} = min_{1 / 2 \leq h \leq 1} Δ (h)$ , then $\frac{8}{\sqrt{5}} Δ_{1} - 8 Δ_{2}$ is equal to (2013 Adv)

Show Answer

Solution:

PLAN As to maximise or minimise area of triangle, we should find area in terms of parametric coordinates and use second derivative test.

Here, tangent at $P (2 \cos θ, \sqrt{3} \sin θ)$ is

$\begin{aligned} \frac{x}{2} \cos θ + \frac{y}{\sqrt{3}} \sin θ = 1 \\ ∴ R (2 \sec θ, 0) \\ \Rightarrow Δ = Area of △ P Q R \\ = \frac{1}{2} (2 \sqrt{3} \sin θ) (2 \sec θ - 2 \cos θ) \\ = 2 \sqrt{3} \cdot \sin^{3} θ / \cos θ \end{aligned}$

When

$\frac{1}{4} \leq \cos θ \leq \frac{1}{2}$

$∴ Δ_{1} = Δ_{max}$ occurs at $\cos θ = \frac{1}{4} = \frac{2 \sqrt{3} \cdot \sin^{3} θ}{\cos θ}$

When $\cos θ = \frac{1}{4} = \frac{45 \sqrt{5}}{8}$

$Δ_{2} = Δ_{min}$ occurs at $\cos θ = \frac{1}{2}$

$= \frac{2 \sqrt{3} \sin^{3} θ}{\cos θ}$

When $\cos θ = \frac{1}{2} = \frac{9}{2}$

$∴ \frac{8}{\sqrt{5}} Δ_{1} - 8 Δ_{2} = 45 - 36 = 9$

पिछला

अगला

Application of Derivatives 4 Question 60

63. A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$ at the points $P$ and $Q$ . If the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ .

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

Application of Derivatives 4 Question 60

63. A vertical line passing through the point (h,0) intersects the ellipse x24+y23=1 at the points P and Q. If the tangents to the ellipse at P and Q meet at the point R.

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

Menu

63. A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$ at the points $P$ and $Q$ . If the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ .