For what values of k does the line y= x + k pass through the circle defined by x^2 + y^2 = 25 at:
a) 2 points
b) 1 point
c) 0 points
sub in y = x+k
x^2 + y^2 = 25
x^2 + (x+k)^2 = 25
x^2 + x^2 + 2kx + k^2 - 25 = 0
2x^2 + 2kx + (k^2-25) = 0
a = 2, b = 2k, c = k^2-25
discriminant = b^2 - 4ac
= 4k^2 - 4(2)(k^2-25)
= 4k^2 - 8k^2 + 100
= 100 - 4k^2
now proceed the same way as in the previous post to "Eloise"
I believe your last few lines are incorrect:
What was written:
4k^2 - 4(2)(k^2-25)
= 4k^2 - 8k^2 + 100
= 100 - 4k^2
What it should read:
4k^2 - 4(2)(k^2-25)
= 4k^2 - 8k^2 + 200
= 200 - 4k^2
To determine the values of \(k\) for which the line \(y = x + k\) passes through the circle \(x^2 + y^2 = 25\), we need to find the intersection points (if any) between the line and the circle.
a) For the line to pass through the circle at two points, the line must intersect the circle at two distinct \(x\) values. Let's solve this algebraically:
Substitute \(y = x + k\) into the equation of the circle:
\[x^2 + (x + k)^2 = 25\]
Expanding and simplifying the equation:
\[2x^2 + 2kx + k^2 - 25 = 0\]
Since we want the line to intersect the circle at two points, the equation must have two distinct solutions. This means the discriminant of the quadratic equation must be positive:
\[\Delta = (2k)^2 - 4(2)(k^2 - 25)\]
Simplifying further:
\[\Delta = 4k^2 - 8k^2 + 200\]
\[\Delta = -4k^2 + 200\]
For the equation to have two distinct solutions, \(\Delta\) must be greater than zero:
\[-4k^2 + 200 > 0\]
Solving this inequality:
\[-4k^2 + 200 > 0\]
\[4k^2 < 200\]
\[k^2 < 50\]
\[-\sqrt{50} < k < \sqrt{50}\]
Therefore, the line \(y = x + k\) passes through the circle at two points for values of \(k\) where \(-\sqrt{50} < k < \sqrt{50}\).
b) For the line to pass through the circle at one point, the line must be a tangent to the circle. This means the quadratic equation must have a discriminant of zero:
\[-4k^2 + 200 = 0\]
Solving for \(k\):
\[4k^2 = 200\]
\[k^2 = \frac{200}{4}\]
\[k^2 = 50\]
\[k = \pm \sqrt{50}\]
Therefore, the line \(y = x + k\) passes through the circle at one point for \(k = \sqrt{50}\) or \(k = -\sqrt{50}\).
c) For the line to not intersect the circle, the discriminant of the quadratic equation must be negative:
\[-4k^2 + 200 < 0\]
Solving this inequality:
\[-4k^2 + 200 < 0\]
\[-4k^2 < -200\]
\[k^2 > 50\]
\[k < -\sqrt{50}\] or \(k > \sqrt{50}\)
Therefore, the line \(y = x + k\) does not pass through the circle for \(k < -\sqrt{50}\) or \(k > \sqrt{50}\).