Find the range of value for $k$ for which $kx + y = 3$ meets $x^2 + y^2 = 5$ in two distinct points.
im so stuck can someone give me a clear guide to the correct method and answer, thank you
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1$\begingroup$ $kx+y=3\implies y=3-kx.$ The line and the circumference have two points in common if $x^2+(3-kx)^2=5$ has two different solutions. $\endgroup$– mflCommented Oct 27, 2014 at 13:28
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4 Answers
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HINT:
Put $y=3-kx$ in the second equation to form a Quadratic Equation in $x$
Each value of $x$ corresponds to the abscissa of the intersection
Do you know how to find the nature of roots of a Quadratic Equation?
answered Oct 27, 2014 at 13:28
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$\begingroup$ Thank you very much :) Once i expand the brackets to $x^2 + 4 - 6kx +kx^2 = 0$ i cant seem to be able to find x still :( $\endgroup$– KellyCommented Oct 27, 2014 at 13:30
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$\begingroup$ @Kelly, You need to arrange as $$Ax^2+Bx+C=0$$ Have you heard of discriminant ? $\endgroup$ Commented Oct 27, 2014 at 13:37
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$\begingroup$ Yes i understand the discriminant and everything, i dont understand where i plug $x$ $\endgroup$– KellyCommented Oct 27, 2014 at 13:39
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$\begingroup$ @Kelly,Why do you want "where I plug " $x$? $\endgroup$ Commented Oct 27, 2014 at 13:40
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$$x^2+y^2=5$$ $$y=3-kx$$
$$x^2+(3-kx)^2=5$$
$$x^2+9+k^2x^2-6kx=5$$ $$(1+k^2)x^2-6kx+4=0$$
Distinct Real roots $\Rightarrow\Delta\gt0$ $$b^2-4ac\gt0$$ where $b=-6k$ , $a=1+k^2$ , $c=4$
$$36k^2-16(1+k^2)>0$$
$$20k^2-16>0$$ $$\Rightarrow |k|\gt\frac{2}{\sqrt5}$$
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$\begingroup$ The discriminant is $b^2 - 4ac$ with $b = -6k$, $a = 1 + k^2$, and $c = 4$. $\endgroup$ Commented Oct 27, 2014 at 13:40
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$\begingroup$ I understand now thank you very much :D $\endgroup$– KellyCommented Oct 27, 2014 at 13:40
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I suggest you to draw a picture, it is easy to see that $-\frac{\sqrt{5}}{2} < k<\frac{\sqrt{5}}{2} $
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This is a circle with diameter 5 centred at the origin, and a line which is pivoted at y=3. Think of the line as a pendulum sweeping through the plane and intersecting with the circle. Since the pivot is above the top of the circle then during its sweep it will intersect at two distinct points with the circle except when the line becomes tangent to the circle at two points. This is when |k|=2/√5. Note that k is the negative of the slope of the line.
