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  • The parabola <img alt="y^2=\lambda x \text { and } 25\left[(x-3)^2+(y+2)^2\right]" src="https://entrancecorner.oncodecogs.com/gif.latex?y%5E2%3D%5Clambda%20x%20%5Ctext%20%7B%20and%20

The parabola y^2=\lambda x \text { and } 25\left[(x-3)^2+(y+2)^2\right] =(3 x-4 y-2)^2 are equal, if \lambda is equal to 

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

6


Answers (1)

best_answer

(d) Let us recall that two parabolas are equal, if the length of their latusrectum are equal.

So, the length of the latusrectum of y^2=\lambda x is \lambda.

The equation of the second parabola is

\begin{aligned} \\25\left\{(x-3)^2+(y+2)^2\right\} =(3 x-4 y-2)^2 \\ \\\Rightarrow \sqrt{(x-3)^2+(y+2)^2}=\frac{|3 x-4 y-2|}{\sqrt{3^2+4^2}} \end{aligned}

Clearly, it represents a parabola having focus at (3,-2) and equation of the directrix as 3 x-4 y-2=0.
\therefore Length of the latusrectum

                               \\=2(\text{distance between focus and directrix})\\ \\=2\left|\frac{3 \times 3-4 \times(-2)-2}{\sqrt{3^2+(-4)^2}}\right|=6

Thus, the two parabolas are equal, if \lambda=6

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