Question #134448
Find the equation of the line perpendicular to 2x+3y=1 and whose distance from the
origin is 7.
1
Expert's answer
2020-10-02T15:57:41-0400

Let the meeting point of the perpendicular lines be (a,b)(a, b)


/OP/=(a2+b2)=7/OP/=\sqrt{(a^2 +b^2)} =7 ..... (i)


a2+b2=49

2x+3y=1 ;y=(23)x+13-(\frac{2}{3}) x+\frac{1} {3}

Gradient =(23)-(\frac{2}{3})


For perpendicular lines,the product of Gradient I and gradient II hence gradient II =32\frac{3}{2}


Gradient=ΔyΔxGradient =\frac{\Delta y} {\Delta x}


32=b0a0\frac{3}{2} =\frac{b-0} {a-0}


2b=3a .... (ii)

Substituting (ii) in (i) ;

a2+(3a2)2=49a^2 +(\frac{3a}{2} )^2 =49


4a2+9a2=1964a^2+9a^2=196


a =4 or -4; b=6 or - 6.


To get the equation, we take the point (4,6) and the gradient 32\frac{3}{2}


32=y6x4\frac{3}{2} =\frac{y-6}{x-4}


2y-12=3x -12


2y=3x


When a=- 4 and b=-6


32=y(6)x(4)\frac{3} {2} =\frac{y-(-6)}{x-(-4)}


3x++ 12=2y++12


2y=3x

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