Question

The centre of the circle passing through the point (0, 1) and touching the curve y = x² at (2, 4) is –

• A. $\left( \frac{- 16}{5},\frac{27}{10} \right)$
• B. $\left( \frac{- 16}{7},\frac{53}{10} \right)$
• C. $\left( \frac{- 16}{5},\frac{53}{10} \right)$
• D. None

Answer 1

Answer: (C)

$\left( \frac{- 16}{5},\frac{53}{10} \right)$

Answer 2

Let the centre be (h, k) then

(h – 0)² + (k – 1)² = (k – 2)² + (k – 4)² (radii² of the same circle)

Ž 4h + 6k = 19 … (i)

Again centre of the circle must lie on the equation of normal to the parabola y = x² at (2, 4).

Thus equation is y – 4 = – $\frac{1}{4}$ (x – 2)

Ž h + 4k = 18

From (i) and (ii) h = – $\frac{16}{5}$, k = $\frac{53}{10}$.