Answer:
It is 3y = 4x + 10
Step-by-step explanation:
Let's first get the slope of the curve.
[ slope is the derivative of the equation ]
[tex] {x}^{2} - xy + {y}^{2} = 7[/tex]
introduce dy/dx :
[tex] \frac{d}{dx} ( {x}^{2} - xy + {y}^{2} ) = \frac{d}{dx} (7) \\ \\ 2x - (y + \frac{dy}{dx} ) + 2y \frac{dy}{dx} = 0[/tex]
make dy/dx the subject:
[tex]2x - y - \frac{dy}{dx} + 2y \frac{dy}{dx} = 0 \\ \\ 2y \frac{dy}{dx} - \frac{dy}{dx} = y - 2x \\ \\ \frac{dy}{dx} (2y - 1) = y - 2x \\ \\ \frac{dy}{dx} = \frac{y - 2x}{2y - 1} [/tex]
At point (-1, 2):
[tex] \frac{dy}{dx} = \frac{2 - 2( - 1)}{2(2) - 1} \\ \\ slope = \frac{4}{3} [/tex]
but a tangent has the same slope as the curve:
[tex]y = mx + c[/tex]
m is the slope
c is the y-intercept
At (-1, 2):
[tex]2 = ( - 1 \times \frac{4}{3} ) + c \\ \\ c = 2 + \frac{4}{3} \\ \\ c = \frac{10}{3} [/tex]
equation:
[tex]y = \frac{4}{3} x + \frac{10}{3} \\ \\ { \boxed{3y = 4x + 10}}[/tex]
