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A function $y = f (x)$ satisfies the following conditions
1) $f (1) = 1$
2) The tangent at any point of the graph of f(x) cuts off on the y-axis an intercept equal to the square of the abscissa of point of tangency.
Which of the following are correct.

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The graph of $y = f (x)$ is passing through (0, 0)

$y = f (x)$ represents a conic whose focus is $(1, \frac{3}{4})$

The area enclosed by $y = f (x)$ and the line $y + 3 = 0$ is $\frac{32}{3}$

$y = f (x)$ has no maxima

answer is A, C, B.

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Detailed Solution

$Y - y = \frac{dy}{dx} (X - x)$
Its Y-intercept is $y - \frac{dy}{dx} x = x^{2}$
$\frac{dy}{dx} - \frac{1}{x} y = - x$

$\frac{y}{x} = - x + c s u b s t i t u t i n g (1, 1) \Rightarrow c = 2$
On solving $y = 2 x - x^{2}$

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