Almost perfectly round

Geometry Level 4

Consider the parabola C 1 : y 2 = 4 x C_{1} : y^{2} = 4x and the circle C 2 : x 2 + y 2 − 6 x + 1 = 0 C_{2} : x^{2} + y^{2} -6x +1 = 0 . Then C 1 C_{1} and C 2 C_{2} :

A) touch each other exactly at two points.
B) touch each other exactly at one point.
C) intersect (but do not touch) at exactly two points.
D) neither intersect nor touch each other.

D D A A B B C C

Tapas Mazumdar by Tapas Mazumdar , 20, India

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1 solution

Michael De Seguin
Sep 25, 2016

Below graph (done using a free online equation sketching) makes this look simple

Let's see if C1 and C2 have intersections/common points. Replacing y 2 y^2 by 4 x 4x in C2, we have:

x 2 + 4 x − 6 x + 1 = 0 x^2 + 4x - 6x + 1 =0 , that is x 2 − 2 x + 1 = 0 x^2 -2x + 1=0 , that is ( x − 1 ) 2 = 0 (x - 1)^2=0

so x = 1 x =1 for which y = ± 2 y =\pm\; 2 in both C1 and C2: one pair of solutions ( 1 2 ) ( 1 − 2 ) { 1 \choose 2}\;{1 \choose-2} and C1 and C2 either intersect or are tangential at these 2 points.

Then let's get the slope of the tangents at these two points for C1 and C2

For C1: f ( x ) = 2 x f(x)= 2\sqrt{x}\; differential f ′ ( x ) = 1 x = 1 f o r x = 1 f'(x) = \frac{1}{\sqrt{x}} = 1\; for\; x=1 = slope tangent at ( 1 2 ) {1 \choose 2}

and: f ( x ) = − 2 x \;f(x)=- 2\sqrt{x}\; differential f ′ ( x ) = − 1 x = − 1 f o r x = 1 f'(x) = \frac{-1}{\sqrt{x}} = -1\; for\; x= 1 = slope tangent at ( 1 − 2 ) {1 \choose -2}

For C2: f ( x ) = − x 2 + 6 x − 1 f(x)= \sqrt{-{x^2}+6x-1}\; differential f ′ ( x ) = − 2 x + 6 2 − x 2 + 6 x − 1 = 1 f o r x = 1 f'(x) = \frac {-2x+6}{2\sqrt{-x^2+6x-1}}\; = 1\; for\; x =1 = slope tangent at ( 1 2 ) {1 \choose 2}

and: f ( x ) = − − x 2 + 6 x − 1 f(x)= - \sqrt{-{x^2}+6x-1}\; differential f ′ ( x ) = 2 x − 6 2 − x 2 + 6 x − 1 = − 1 f o r x = 1 f'(x) = \frac {2x-6}{2\sqrt{-x^2+6x-1}}\; = - 1\; for\; x =1 = slope tangent at ( 1 − 2 ) {1 \choose -2}

C1 and C2 both go thru the points ( 1 2 ) ( 1 − 2 ) {1 \choose 2}\;{1 \choose-2} and have the same tangents at these two points therefore C 1 a n d C 2 a r e a l s o t a n g e n t a t t h e s e 2 p o i n t s \boxed{C1\; and \; C2\; are\; also\; tangent\; at\; these\; 2\; points} (answer A)

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