JEE Main 2020 — Parabola Question with Solution

Circle passes through A(0, 1) and B(2, 4).

y = x²

$$\Rightarrow$$ $${\frac{dy}{dx}|}_B$$ = 4

tangent at (2,4) is

(y – 4) = 4(x – 2)

4x – y – 4 = 0

Equation of circle

(x - 2)² + (y–4)² + $$\lambda$$(4x–y - 4) = 0

Passing through (0,1)

$$\therefore$$ 4 + 9 + $$\lambda$$(–5) = 0

$$\Rightarrow$$ $$\lambda$$ = $$\frac{13}{5}$$

$$\therefore$$ Circle is

x²– 4x + 4 + y² – 8y + 16 + $$\frac{13}{5}$$[4x - y - 4] = 0

$$\Rightarrow$$ x² + y² + $$\left(\frac{52}{5}-4\right)$$x - $$\left(8+\frac{13}{5}\right)$$y + 20 - $$\frac{52}{5}$$ = 0

$$\Rightarrow$$ x² + y² + $$\frac{32}{5}x-\frac{53}{5}y$$ + $$\frac{48}{5}$$ = 0

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