Question
Question: The centre of the circle passing through the point (0, 1) and touching the curve y = x<sup>2</sup> a...
The centre of the circle passing through the point (0, 1) and touching the curve y = x2 at (2, 4) is
A
(-16/5, 27/10)
B
(-16/7, 5/10)
C
(-16/5, 53/10)
D
None of these
Answer
(-16/5, 53/10)
Explanation
Solution
Tangent to the parabola y = x2 at (2, 4) is
21(y+4)=x.26muor6mu4x−y−4=0
It is also a tangent to the circle so that the centre lies on the normal through (2, 4) whose equation is x+4y = λ, where
2 + 16 = λ.
∴ x+4y = 18 is the normal on which lies (h, k).
∴ h+4k = 18 ..(i)
Again distance of centre (h, k) from (2, 4) and (0, 1) on the circle are equal.
∴ (h-2)2 + (k-4)2 = h2 + (k –1)2
∴ 4h+6k = 19 ..(ii)
Solving (i) and (ii), we get the centre =(-16/5, 53/10)