SATHEE: Application of Derivatives 1 Question 10

Application of Derivatives 1 Question 10

####10. The normal to the curve $x^{2} + 2 x y - 3 y^{2} = 0$ at $(1, 1)$

(a) does not meet the curve again

(b) meets in the curve again the second quadrant

(d) meets the curve again in the fourth quadrant

(2015 Main)

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Answer:

Correct Answer: 10. $a = - \frac{1}{2}, b = - \frac{3}{4}, c = 3$

Solution:

Given equation of curve is

$x^{2} + 2 x y - 3 y^{2} = 0$

On differentiating w.r.t $x$ , we get

$\begin{aligned} 2 x + 2 x y^{'} + 2 y - 6 y y^{'} = 0 \Rightarrow y^{'} = \frac{x + y}{3 y - x} \\ At x = 1, y = 1, y^{'} = 1 \\ i.e. {\frac{d y}{d x}}_{(1, 1)} = 1 \end{aligned}$

Equation of normal at $(1, 1)$ is

$\begin{aligned} y - 1 = - \frac{1}{1} (x - 1) \Rightarrow & y - 1 = - (x - 1) \\ \Rightarrow & x + y = 2 \end{aligned}$

On solving Eqs. (i) and (ii) simultaneously, we get

$\begin{array}{cc} \Rightarrow & x^{2} + 2 x (2 - x) - 33 (2 - x)^{2} = 0 \\ \Rightarrow & x^{2} + 4 x - 2 x^{2} - 3 (4 + x^{2} - 4 x) = 0 \\ \Rightarrow & - x^{2} + 4 x - 12 - 3 x^{2} + 12 x = 0 \\ \Rightarrow & - 4 x^{2} + 16 x - 12 = 0 \\ \Rightarrow & 4 x^{2} - 16 x + 12 = 0 \\ \Rightarrow & x^{2} - 4 x + 3 = 0 \\ \Rightarrow & (1 - 1) (x - 3) = 0 \\ ∴ & x = 1, 3 \end{array}$

Now, when $x = 1$ , then $y = 1$

and when $x = 3$ , then $y = - 1$

$∴ P = (1, 1) and Q = (3, - 1)$

Hence, normal meets the curve again at $(3, - 1)$ in fourth quadrant.

Alternate Solution

$\begin{aligned} Given, & x^{2} + 2 x y - 3 y^{2} & = 0 \\ \Rightarrow & (x - y) (x + 3 y) & = 0 \\ \Rightarrow & x - y = 0 or x + 3 y & = 0 \end{aligned}$

Equation of normal at $(1, 1)$ is

$y - 1 = - 1 (x - 1) \Rightarrow x + y - 2 = 0$

It intersects $x + 3 y = 0$ at $(3, - 1)$ and hence normal meets the curve in fourth quadrant.

Application of Derivatives 1 Question 10

Answer:

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Application of Derivatives 1 Question 10

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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