Conic Sections Question 60
Question: If $ a\ne 0 $ and the line $ 2bx+3cy+4d=0 $ passes through the points of intersection of the parabolas $ y^{2}=4ax $ and $ x^{2}=4ay $ , then
[AIEEE 2004]
Options:
A) $ d^{2}+{{(3b-2c)}^{2}}=0 $
B) $ d^{2}+{{(3b+2c)}^{2}}=0 $
C) $ d^{2}+{{(2b-3c)}^{2}}=0 $
D) $ d^{2}+{{(2b+3c)}^{2}}=0 $
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Answer:
Correct Answer: D
Solution:
Given prarbolas are $ y^{2}=4ax $ ……(i) $ x^{2}=4ay $ ……(ii) Putting the value of y from (ii) in (i), we get $ \frac{x^{4}}{16a^{2}}=4ax\Rightarrow x(x^{3}-64a^{3})=0\Rightarrow x=0,4a $ . from (ii), $ y=0,4a $ . Let $ A\equiv (0,0);B\equiv (4a,4a) $
Since, given line $ 2bx+3cy+4d=0 $ passes through A and B,$ d=0 $ and $ 8ab+12ac=0\Rightarrow 2b+3c=0 $ ,( $ \because $
$ a\ne 0 $ ) Obviously, $ d^{2}+{{(2b+3c)}^{2}}=0 $ .
BETA
