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M17 #12

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Manager
Joined: 25 Jul 2010
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M17 #12 [#permalink]  09 Oct 2010, 00:53
Is line Y = KX + B tangent to circle $$X^2 + Y^2 = 1$$ ?

1. K + B = 1
2. $$K^2 + B^2 = 1$$

A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D) EACH statement ALONE is sufficient
E) Statements (1) and (2) TOGETHER are NOT sufficient
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Re: M17 #12 [#permalink]  09 Oct 2010, 01:05
1
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Orange08 wrote:
Is line Y = KX + B tangent to circle $$X^2 + Y^2 = 1$$ ?

1. K + B = 1
2. $$K^2 + B^2 = 1$$

A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D) EACH statement ALONE is sufficient
E) Statements (1) and (2) TOGETHER are NOT sufficient

Lets solve the equations simultaneously. If a line is a tangent to the circle the equations will have exactly one solution which is the point of intersection.

$$y=kx+b$$
$$x^2+y^2=1$$
$$x^2+(kx+b)^2=1$$
$$(1+k^2)x^2+2kbx+b^2-1=0$$

For a quadratic equation to have a single solution, the discriminant (b^2-4ac) must be 0. Therefore :

$$(2kb)^2 - 4(1+k^2)(b^2-1)=0$$
$$4k^2b^2 - 4(b^2-1+k^2b^2-k^2)=0$$
$$-4b^2+4+4k^2=0$$
$$b^2-k^2=1$$

(1)k+b=1. Insufficient to show $$b^2-k^2=1$$
(2)$$b^2+k^2=1$$, Insufficient to show $$b^2-k^2=1$$
(1+2) Both together are also insufficient

You can also refer to coordinate-geometry-69516.html for an example based approach
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Re: M17 #12 [#permalink]  09 Oct 2010, 01:18
super. Many thanks.
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Re: M17 #12 [#permalink]  09 Oct 2010, 01:29
Expert's post
Orange08 wrote:
Is line Y = KX + B tangent to circle $$X^2 + Y^2 = 1$$ ?

1. K + B = 1
2. $$K^2 + B^2 = 1$$

The circle represented by the equation $$x^2+y^2 = 1$$ is centered at the origin and has the radius of $$r=\sqrt{1}=1$$ (for more on this check Coordinate Geometry chapter of math book: math-coordinate-geometry-87652.html ).

(1) K + B = 1 --> if $$k=0$$ and $$b=1$$ then the equation of the line becomes $$y=1$$ and this line is tangent to the circle but if $$k=1$$ and $$b=0$$ then th equation of the line becomes $$y=x$$ and this line is NOT tangent to the circle. Not sufficient.

(2) K^2 + B^2 = 1. The same example is valid for this statement too. Not sufficient.

(1)+(2) Again the same example satisfies both statement: if $$k=0$$ and $$b=1$$ then the equation of the line becomes $$y=1$$ and this line is tangent to the circle but if $$k=1$$ and $$b=0$$ then th equation of the line becomes $$y=x$$ and this line is NOT tangent to the circle. Not sufficient.

Attachment:

graph.php.png [ 15.97 KiB | Viewed 593 times ]

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Re: M17 #12   [#permalink] 09 Oct 2010, 01:29
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M17 #12

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