If the parabolas25 x -32- y -22-3 x -4 y -22- y 2- xare equal- then - isA- 1B- 3C- 12D- 6

The correct option is D 6Parabolas are equal if length of their latus rectam nbsp;are equal ∴ (x 3)^2+(y+2)^2=(3x 4y 2)^225 ⇒√((x 3)^2+(y+2)^2)=|3x 4y 2|5 ∴ ...

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Standard XII

Mathematics

Latus Rectum

If the parabo...

Question

If the parabolas
$25 {(x - 3)^{2} + (y + 2)^{2}} = (3 x - 4 y - 2)^{2}, y^{2} = λ x$
are equal, then $λ$ is

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Solution

The correct option is C $6$
Parabolas are equal if length of their latus rectam are equal

$∴ (x - 3)^{2} + (y + 2)^{2} = \frac{(3 x - 4 y - 2)^{2}}{25} \Rightarrow \sqrt{(x - 3)^{2} + (y + 2)^{2}} = \frac{| 3 x - 4 y - 2 |}{5}$
$∴$ LLR $= 2 Distance between focus and directrix = 2 ∣ ∣ ∣ \begin{matrix} \frac{3 \times 3 - 4 \times (- 2) - 2}{\sqrt{3^{2} + (- 4)^{2}}} \end{matrix} ∣ ∣ ∣ = 6$
Also LLR of $y^{2} = λ x$ is $λ$
$∴ λ = 6$

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If the parabolas 25{(x−3)2+(y+2)2}=(3x−4y−2)2,y2=λx are equal, then λ is

The correct option is C 6Parabolas are equal if length of their latus rectam are equal ∴(x−3)2+(y+2)2=(3x−4y−2)225⇒√(x−3)2+(y+2)2=|3x−4y−2|5 ∴ LLR=2 Distance between focus and directrix=2∣∣∣3×3−4×(−2)−2√32+(−4)2∣∣∣=6 Also LLR of y2=λx is λ ∴λ=6

If the parabolas
$25 {(x - 3)^{2} + (y + 2)^{2}} = (3 x - 4 y - 2)^{2}, y^{2} = λ x$
are equal, then $λ$ is