find the equation of straight line paing through (1,-1)and perpendicular to x2 +xy+6y2.
Hello,
1. I assume that the equation of the curve reads:
$\displaystyle x^2+xy+6y^2=6$ The curve is an ellipse which contains the point A(1, -1).
2. Use implicite deifferentiation to get the gradient at A:
$\displaystyle 2x+(y+x \cdot y') +12y \cdot y'=0$ . Plug in the values for x and y and solve the equation for y': I've got $\displaystyle y'=\frac1{11}$
The direction perpendicular to y' has the slope m = -11. Use the point-slope-formula to calculate the equation of the line:
$\displaystyle \frac{y-(-1)}{x-1}=-11~\iff~\boxed{y=-11x + 10}$
I've attached a sketch of the situation.