Tricky Question - challenge

**thejoester** · December 5th 2006, 07:38 PM

my teacher said anyone in my class who gets this question would get a prize in front of the whole school.... okay.... i exaggerate.... but still, i want to understand this.

the question is:

consider the circle x^2 + y^2 = 25 and the line y = x +k, where k is any real number. Determine the values of k for which the line will intersect the circle in one, two, or no points. Repeat for the circle x^2 + y^2 = 49 and the line y = x+k.

generalize your results for a circle of radius r and the line y = x+k

Note: the question asks for you to do two. Dont worry about it, just explain the steps and i will understand it.

Thanks in advance

**ThePerfectHacker** · December 5th 2006, 07:57 PM

Originally Posted by thejoester

consider the circle x^2 + y^2 = 25 and the line y = x +k, where k is any real number. Determine the values of k for which the line will intersect the circle in one, two, or no points. Repeat for the circle x^2 + y^2 = 49 and the line y = x+k.

What class is this?

Okay,
If they intersect we can substitute $y=x+k$ into the equation,
$x^2+(x+k)^2=25$
$x^2+x^2+2xk+k^2=25$
$2x^2+2xk+k^2=25$
$2x^2+2xk+(k^2-25)=0$
To have 2 distinct points we need the discrimant to be positive.
$(2k)^2-4(2)(k^2-25)>0$
Thus,
$4k^2-8k^2+200>0$
$200-4k^2>0$
$50-k^2>0$
$50>k^2$
$k^2<50$
$|k|<\sqrt{50}$

**Soroban** · December 6th 2006, 08:37 AM

Hello, thejoester!

Consider the circle $x^2 + y^2 \,= \,25$ and the line $y \,= \,x +k$
Determine the values of $k$ for which the line will intersect the circle in 1, 2, or 0 points.

To complete what ThePerfectHacker started . . .

The discriminant is: . $D \:=\:4(50 - k^2)$

To have one point of intersection, $D = 0$
. . $4(50 - k^2)\:=\:0\quad\Rightarrow\quad k^2 \,=\,50\quad\Rightarrow\quad k \,=\,\pm5\sqrt{2}$

To have two points of intersection: $D > 0$
. . $4(50-k^2)\:>\:0\quad\Rightarrow\quad k^2 \:<\:50\quad\Rightarrow\quad |k| \:< \:5\sqrt{2}$

To have no points of intersection: $D < 0$
. . $4(50-k^2) \:<\:0\quad\Rightarrow\quad k^2 \:>\:50\quad\Rightarrow\quad|k|\:>\:5\sqrt{2}$

The "graph" looks like this:

. . $\overbrace{- - - }^{\text{0 points}}+\overbrace{ - - - + - - - }^{\text{2 points}}+\overbrace{ - - -}^{\text{0 points}}$
. . . . . $\text{-}5\sqrt{2}$ . . . . . . . . . $5\sqrt{2}$
. . . . . $^{\text{1 point}}$ . . . . . . . . . $^{\text{1 point}}$

**thejoester** · December 6th 2006, 02:28 PM

thanks a lot guys, i really appreciate it.
i have one more question though, would either of you know how to graph this on excel?

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