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Line passing through mid point the point (0, -4) and (-4, 4) and slope is same as slope of function 2x2 + y2 + 4x = 0 at point (1,1) is given by
1. y = 4x - 8
2. - y = - 8x - 4
3. - y = 4x - 8
4. y = - 4x - 8

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Correct Answer - Option 4 : y = - 4x - 8

Concept:

Equation of a line: Equation of line passing through (x1, y1) with slope m is given by

(y - y1) = m(x - x1)

Mid-Point formula: If two-point A and B are such that, A ((x1, y1) and B(x2, y2), then coordinate of midpoint of A & B is given by 

\((\frac{x_1\ +\ x_2}{2},\ \frac{\ y_1\ +\ y_2}{2})\)

Slope of  function: Slope of function y = f(x) is given by 

\(\frac{dy}{dx}\ =\ f'(x)\)

Calculation:

Given that,

2x2 + y2 + 4x = 0 

Differentating both side with respect to x,

\(⇒ \ \frac{d}{dx}(2x^2 + y^2 + 4x) = 0 \)

\(⇒ \ (4x + 2y\frac{dy}{dx} + 4) = 0 \)             (\(\frac{d}{dx}x^n\ =\ nx^{n\ -\ 1}\))

\(⇒ \ \frac{dy}{dx} \ =\ \frac{-4x\ -\ 4}{2y}\)      

Therefore, the slope of the function at the point (1, 1)

\(⇒ \ \frac{dy}{dx} \ =\ -4\)                          ........(1)

Mid-point of (0, -4) and (-4, 4) is

\((\frac{0\ +\ (-4)}{2},\ \frac{\ (-4 \ +\ 4)}{2})\) =  (-2, 0)       (by using mid-point formula)

Therefore, the equation of passing through the point (-2, 0) and slope -4 by using the relation

(y - y1) = m(x - x1

(y - 0) = -4(x + 2)

⇒ y = -4x - 8

Hence, option 4 is correct.

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