Q. 184.8( 4 Votes )

At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?

Answer :

Given: equation of a curve x2 + y2 – 2x – 4y + 1 = 0


To find: the points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis


Explanation:


Now given equation of curve as x2 + y2 – 2x – 4y + 1 = 0


Differentiating this with respect to x, we get



Applying the sum rule of differentiation, we get







As it is given the tangents are parallel to the y-axis,


Hence


Substituting the value from equation (i), we get



y-2 = 0


y = 2


Now substituting y = 2 in curve equation, we get


x2 + y2 – 2x – 4y + 1 = 0


x2 + 22 – 2x – 4(2) + 1 = 0


x2 + 4 – 2x – 8 + 1 = 0


x2– 2x-3 = 0


Now splitting the middle term, we get


x2-3x+x-3 = 0


x(x-3)+1(x-3) = 0


(x+1)(x-3) = 0


x+1 = 0 or x-3 = 0


x = -1 or x = 3


So the required points are (-1, 2) and (3, 2).


Hence the points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis are (-1, 2) and (3, 2).


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