SATHEE: Ellipse 1 Question 10

Ellipse 1 Question 10

11. Let $P$ be a variable point on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with foci $F_{1}$ and $F_{2}$ . If $A$ is the area of the $△ P F_{1} F_{2}$ , then the maximum value of $A$ is… .

(1994, 2M)

Analytical & Descriptive Question

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Answer:

Correct Answer: 11. $\frac{x^{2}}{a^{2}} + \frac{y^{2} (r + s)^{2}}{(a r + b s)^{2}} = 1$

Solution:

Given,

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

Foci $F_{1}$ and $F_{2}$ are $(- a e, 0)$ and $(a e, 0)$ , respectively. Let $P (x, y)$ be any variable point on the ellipse.

The area $A$ of the triangle $P F_{1} F_{2}$ is given by

$\begin{aligned} A & = \frac{1}{2} | \begin{array}{ccc} x & y & 1 \\ - a e & 0 & 1 \\ a e & 0 & 1 \end{array} | \\ = \frac{1}{2} (- y) (- a e \times 1 - a e \times 1) \end{aligned}$

$= - \frac{1}{2} y (- 2 a e) = a ey = a e \cdot b \sqrt{1 - \frac{x^{2}}{a^{2}}}$

So, $A$ is maximum when $x = 0$ .

$∴$ Maximum of $A = a b e = a b \sqrt{1 - \frac{b^{2}}{a^{2}}} = a b \sqrt{\frac{a^{2} - b^{2}}{a^{2}}}$

$= b \sqrt{a^{2} - b^{2}}$

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Ellipse 1 Question 10

11. Let $P$ be a variable point on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with foci $F_{1}$ and $F_{2}$ . If $A$ is the area of the $△ P F_{1} F_{2}$ , then the maximum value of $A$ is… .

Analytical & Descriptive Question

Answer:

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Ellipse 1 Question 10

11. Let P be a variable point on the ellipse x2a2+y2b2=1 with foci F1 and F2. If A is the area of the △PF1F2, then the maximum value of A is… .

Analytical & Descriptive Question

Answer:

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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11. Let $P$ be a variable point on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with foci $F_{1}$ and $F_{2}$ . If $A$ is the area of the $△ P F_{1} F_{2}$ , then the maximum value of $A$ is… .