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.