Question

Solve the simultaneous equation y=8-x 2x/2 +xy=-16

Answer 1

Answer:

state,giving reason for your answer,whether the line y=8-x is a tangent to the curve 2xsquared + xy = -16

y = 8 - x

2x^2 + xy = -16

Substitute the 1st equation, y = 8 - x, into the 2nd equation,

2x^2 + x(8 - x) = -16

2x^2 + 8x - x^2 = -16

x^2 + 8x + 16 = 0

(x + 4)^2 = 0

x = -4 (twice)

There is a double root to the quadratic equation, i.e. x = -4. The same root occurs twice meaning that the straight line, y = 8 - x, only touches the curve 2x^2 +xy = -16 in one point only (It touches the same point twice).

Since this is a straight line that touches a curve at one point only, then that straight line is a tangent to the curve (at the point of touching).