Maths-
General
Easy
Question
Consider the two curves C1 : y2 = 4xC2 : x2 + y2 – 6x + 1 = 0Then,
- C1 and C2 touch each other only at one point
- C1 and C2 touch each other exactly at two points
- C1 and C2 intersect (but do not touch) at exactly two points
- C1 and C2 neither intersect nor touch each other
The correct answer is: C1 and C2 touch each other exactly at two points
To find the correct option.
Solving the two equations, we get x=1 and y=±2
So the two curves meet at two points (1,2) and (1,−2).
Equation of the tangent at (1,2) to C1 is y(2)=2(x+1)
y=x+1
and Equation of the tangent at (1,2) to C2 is
x.1+y(2)−3(x+1)+1=0
y=x+1
Showing that C1 and C2 have a common tangent at the point (1,2).
Similarly they have a common tangent
y=−(x+1)at(1,−2)
So the two curves meet at two points (1,2) and (1,−2).
Equation of the tangent at (1,2) to C1 is y(2)=2(x+1)
y=x+1
and Equation of the tangent at (1,2) to C2 is
x.1+y(2)−3(x+1)+1=0
Similarly they have a common tangent
y=−(x+1)at(1,−2)
Hence, the two curves touch each other exactly at two points.
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