ISI2018-MMA-21

Question

The angle between the tangents drawn from the point $(1, 4)$ to the parabola $y^2 = 4x$ is

• A. $\pi /2$
• B. $\pi /3$
• C. $\pi /4$
• D. $\pi /6$

Answer 1

Answer: B

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Let $P(t^2,2t)$ be the parametric point on the parabola. Then the slope of the parabola at $P$ is given by

$$\frac{dy}{dx}=\frac{1}{t}$$

Now, the slope of the tangent at $P$ passing through the point $P$ and $(1,4)$ is given by $$\frac{2t-4}{t^2-1}$$.

Hence, equating the slopes at point $P$, we have the following $$\frac{1}{t}=\frac{2t-4}{t^2-1}$$

The roots of the equation are $t_1$ and $t_2$.

The angle between the two tangents is given by

$$\tan\alpha =\left|\frac{t_2-t_1}{1+t_1t_2}\right|$$

On solving, we will get $\alpha=\frac{\pi}{3}$.

Comments

isn’t this (Pi / 4)??