SATHEE: Ellipse 2 Question 2

Ellipse 2 Question 2

2. The tangent and normal to the ellipse $3 x^{2} + 5 y^{2} = 32$ at the point $P (2, 2)$ meets the $X$ -axis at $Q$ and $R$ , respectively. Then, the area (in sq units) of the $△ P Q R$ is

(a) $\frac{16}{3}$

(b) $\frac{14}{3}$

(d) $\frac{68}{15}$

(2019 Main, 10 April II)

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Solution:

Equation of given ellipse is

$3 x^{2} + 5 y^{2} = 32$

Now, the slope of tangent and normal at point $P (2, 2)$ to the ellipse (i) are respectively

$m_{T} = {\frac{d y}{d x} |}_{(2, 2)} and m_{N} = - {\frac{d x}{d y} |}_{(2, 2)}$

On differentiating ellipse (i), w.r.t. $x$ , we get

$6 x + 10 y \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{3 x}{5 y}$

So, $m_{T} = - {\frac{3 x}{5 y} |}_{(2, 2)} = - \frac{3}{5}$ and $m_{N} = {\frac{5 y}{3 y} |}_{(2, 2)} = \frac{5}{3}$

Now, equation of tangent and normal to the given ellipse (i) at point $P (2, 2)$ are

$(y - 2) = - \frac{3}{5} (x - 2)$

and $(y - 2) = \frac{5}{3} (x - 2)$ respectively.

It is given that point of intersection of tangent and normal are $Q$ and $R$ at $X$ -axis respectively.

So, $Q \frac{16}{3}, 0$ and $R \frac{4}{5}, 0$

$∴$ Area of $△ P Q R = \frac{1}{2} (Q R) \times$ height

$\begin{matrix} = \frac{1}{2} \times \frac{68}{15} \times 2 = \frac{68}{15} sq units \\ [∵ Q R = \sqrt{\frac{16}{3} - {\frac{4}{5}}^{2}} = \sqrt{\frac{68}{15}} = \frac{68}{15} and height = 2] \end{matrix}$

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Ellipse 2 Question 2

2. The tangent and normal to the ellipse $3 x^{2} + 5 y^{2} = 32$ at the point $P (2, 2)$ meets the $X$ -axis at $Q$ and $R$ , respectively. Then, the area (in sq units) of the $△ P Q R$ is

Solution:

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Ellipse 2 Question 2

2. The tangent and normal to the ellipse 3x2+5y2=32 at the point P(2,2) meets the X-axis at Q and R, respectively. Then, the area (in sq units) of the △PQR 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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2. The tangent and normal to the ellipse $3 x^{2} + 5 y^{2} = 32$ at the point $P (2, 2)$ meets the $X$ -axis at $Q$ and $R$ , respectively. Then, the area (in sq units) of the $△ P Q R$ is