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451
MathematicsDifficultMCQAP EAMCET · 2022
शांकव $25[(x-2)^2+(y-3)^2]=(3x-4y+7)^2$ के नाभिलंब की लंबाई है
A
$\frac{1}{5}$
B
$\frac{2}{5}$
C
$\frac{3}{5}$
D
$\frac{4}{5}$
Solution
(D) दिया गया समीकरण $25[(x-2)^2+(y-3)^2]=(3x-4y+7)^2$ है।
$25$ से भाग देने पर,हमें $(x-2)^2+(y-3)^2 = \left(\frac{3x-4y+7}{5}\right)^2$ प्राप्त होता है।
यह $SP^2 = PM^2$ के रूप में है,जहाँ $S(2,3)$ नाभि है और $3x-4y+7=0$ नियता है।
यहाँ,नाभि से नियता की दूरी $a = \left|\frac{3(2)-4(3)+7}{\sqrt{3^2+(-4)^2}}\right| = \left|\frac{6-12+7}{5}\right| = \frac{1}{5}$ है।
परवलय के नाभिलंब की लंबाई $4a$ होती है।
अतः,लंबाई $= 4 \times \frac{1}{5} = \frac{4}{5}$.
$25$ से भाग देने पर,हमें $(x-2)^2+(y-3)^2 = \left(\frac{3x-4y+7}{5}\right)^2$ प्राप्त होता है।
यह $SP^2 = PM^2$ के रूप में है,जहाँ $S(2,3)$ नाभि है और $3x-4y+7=0$ नियता है।
यहाँ,नाभि से नियता की दूरी $a = \left|\frac{3(2)-4(3)+7}{\sqrt{3^2+(-4)^2}}\right| = \left|\frac{6-12+7}{5}\right| = \frac{1}{5}$ है।
परवलय के नाभिलंब की लंबाई $4a$ होती है।
अतः,लंबाई $= 4 \times \frac{1}{5} = \frac{4}{5}$.
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