The equation of the curve whose tangent at any point (x,y) makes an angle tan−1(2x+3y) with x-axis and which passes through (1,2) is
A
6x+9y+2=26e3(x−1)
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B
6x−9y+2=26e3(x−1)
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C
6x+9y−2=26e3(x−1)
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D
6x−9y−2=26e3(x−1)
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Solution
The correct option is A6x+9y+2=26e3(x−1) Given dydx=(2x+3y) Let 2x+3y=z ⇒2+3dydx=dzdx⇒13(dzdx−2)=dydx ⇒13(dzdx−2)=z⇒dzdx=3z+2 ⇒∫dz3z+2=∫dx⇒log(3z+2)=3x+c ⇒∫dz3z+2=∫dx⇒log(3z+2)=3x+c ⇒3z+2=e3x+c ⇒6x+9y+2=e3x+c...(1) 6+18+2=e3x+c 26=e3+c⇒ec=26.e−3 Then the equation is 6x+9y+2=26e3(x−1)