Please help me with this proof that involves quadratic equations.?

Question

Show that the straight line x+y = k will intersect the curve
x^2 - 6x + y^2 - 8y + 4 = 0 at two distinct points
if k^2 - 14k + 7 <0.

Step by step , please.

This is not homework. I am learning what I should learn next year. (what's the verb for it?) I am being hard-working.

Thank you very much, HY. I'll check up tomorrow and give you that 10 points.

Answer 1

I'll tell you the steps, and you do them:

Make y the subject of the first equation
y = ...

and then substitute this expression for y in the second equation.

Expand and collect terms, so you have a quadratic equation in x, with some of the coefficients involving k.

The condition for the line to cut the curve in two distinct points is for this quadratic equation to have two distinct roots. The condition for that is
b^2 - 4ac > 0

Hint: When you work out b^2 - 4ac it will have -k^2 in it. To get k^2 you multiply both sides by -1, and that's why > has to be reversed to give <

If you try this and still have trouble you can email me
[e-mail]

Answer 2

just to add on to what Hy has said, if the line touches the curve at only one point, b^2-4ac = 0. If the line does not cut the curve, b^2-4ac<0