Parabola 2 Question 18

18. The point of intersection of the tangents at the ends of the latusrectum of the parabola $y^{2}=4 x$ is … .

(1994, 2M)

Analytical & Descriptive Questions

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Solution:

  1. The coordinates of extremities of the latusrectum of $y^{2}=4 x$ are $(1,2)$ and $(1,-2)$.

Equations of tangents at these points are

$$ \begin{array}{rlrl} & y \cdot 2=\frac{4(x+1)}{2} \Rightarrow 2 y & =2(x+1) \\ \text { and } & & y(-2) & =\frac{4(x+1)}{2} \\ \Rightarrow & & -2 y & =2(x+1) \end{array} $$

The point of intersection of these tangents can be obtained by solving Eqs. (i) and (ii) simultaneously.

$$ \begin{aligned} \therefore & & -2(x+1) & =2(x+1) \\ \Rightarrow & & 0 & =4(x+1) \\ \Rightarrow & & -1 & =x \Rightarrow y=0 \end{aligned} $$

Therefore, the required point is $(-1,0)$.



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