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  • Consider the two curves <img alt="\mathrm{\begin{aligned} & C_1: y^2=4 x \\ & C_2: x^2+y^2-6 x+1=0 . \text { Then, } \end{aligned}}" src="https://entrancecorner.oncodecogs.com/gif.la

Consider the two curves
\mathrm{\begin{aligned} & C_1: y^2=4 x \\ & C_2: x^2+y^2-6 x+1=0 . \text { Then, } \end{aligned}}
 

Option: 1

\mathrm{C_1 \text { and } C_2} touch each other only at one point 


Option: 2

\mathrm{C_1 \text { and } C_2} touch each other exactly at two points 


Option: 3

\mathrm{C_1 \text { and } C_2} intersect (but do not touch) at exactly two points 


Option: 4

\mathrm{C_1 \text { and } C_2} neither intersect nor touch each other. 


Answers (1)

best_answer

\mathrm{\text { Solving } y^2=4 x \text { with } x^2+y^2-6 x+1=0}

\mathrm{\text { We have } x^2+4 x-6 x+1=0 \Rightarrow x^2-2 x+1=0}

\mathrm{\Rightarrow(x-1)^2=0 \Rightarrow x=1,1 .}

As the equation has a double root, the two curves \mathrm{C_1} and \mathrm{C_2} touch each other at x = 1. To find out at how many points, we have to find the value of y corresponding to

\mathrm{x=1, y^2=4 x \text { gives } y^2=4 \Rightarrow y=-2,2}

Then \mathrm{C_1} and \mathrm{C_2} touch each other at exactly two points (1, – 2) and (1, 2).

Posted by

shivangi.bhatnagar

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