The equation of line belonging to the family of lines $$p(2x+3y$$−$$13)+q(x$$−$$y+1)=0$$ , and which is at maximum distance from the orioin is

Question

The equation of line belonging to the family of lines $$p(2x+3y$$−$$13)+q(x$$−$$y+1)=0$$ ,  and which is at maximum distance from the orioin is

Infinity Learn · Accepted Answer

Given family of lines $$p(2x+3y$$−$$13)+q(x$$−$$y+1)=0$$⇒$$2x+3y$$−$$13+qp(x$$−$$y+1)=0$$⇒$$2x+3y$$−$$13+$$λ(x$$−$$y+1)=0$$∵ Let $$pq=$$λ⇒$$x(2+$$λ$$)+y(3$$−λ$$)+$$λ−$$13=0$$………...$$(1) The distance from origin to (1) is $$D=$$|λ−$$13$$|(2+$$λ$$)2+(3$$−λ$$)2=$$|λ−$$13$$|$$2$$λ$$2$$−$$2$$λ$$+13$$ Differentiate with respect to ′λ′⇒dDdλ$$=2$$λ$$2$$−$$2$$λ$$+13$$−$$($$λ−$$13)(4$$λ−$$2)22$$λ$$2$$−$$2$$λ$$+132$$λ$$2$$−$$2$$λ$$+13=2$$λ$$2$$−$$2$$λ$$+13$$−$$($$λ−$$13)(2$$λ−$$1)2$$λ$$2$$−$$2$$λ$$+1332For$$ maximum value of D ,dDdλ$$=0$$⇒$$2$$λ$$2$$−$$2$$λ$$+13=($$λ−$$13)(2$$λ−$$1)⇒λ$$=0$$ Substituting λ$$=0$$ in equation (1) we get $$2x+3y$$−$$13=0$$

$The equation of line belonging to the family of lines p (2 x + 3 y - 13) + q (x - y + 1) = 0,$
$and which is at maximum distance from the orioin is$

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The equation of line belonging to the family of lines p(2x+3y−13)+q(x−y+1)=0 , and which is at maximum distance from the orioin is

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$The equation of line belonging to the family of lines p (2 x + 3 y - 13) + q (x - y + 1) = 0,$
$and which is at maximum distance from the orioin is$