line through the point A (2$$, $$4) intersects the line x $$+$$ y $$=$$ $$9$$ at the point P. The minimum distance of AP, is

Question

line through the point A (2$$, $$4) intersects the line x $$+$$ y $$=$$ $$9$$ at the point P. The minimum distance of AP, is

Infinity Learn · Accepted Answer

The equation of a line passing through $$A(2$$, $$4)$$ is x−$$2cos$$⁡θ$$=y$$−$$4sin$$⁡θSuppose it cuts the line x $$+$$ y $$=$$ $$9$$ at point P whose coordinates are given by  x−$$2cos$$⁡θ$$=y$$−$$4sin$$⁡θ$$=r$$ i.e. $$x=2+rcos$$⁡θ,$$y=4+rsin$$⁡θ∴    $$2+rcos$$⁡θ$$+4+rsin$$⁡θ$$=9$$⇒    $$r($$$$cos$$⁡θ$$+$$$$sin$$⁡θ$$)=3$$⇒    $$r=3cos$$⁡θ$$+$$$$sin$$⁡θ⇒ r≥$$32$$ ∵$$cos$$⁡θ$$+$$$$sin$$⁡θ≤$$2$$∴$$1cos$$⁡θ$$+$$$$sin$$⁡θ≥$$12Hence$$, the minimum value of AP is $$32$$

line through the point $A (2, 4)$ intersects the line $x + y = 9$ at the point $P .$ The minimum distance of $A P,$ is

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line through the point A (2, 4) intersects the line x + y = 9 at the point P. The minimum distance of AP, is

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line through the point $A (2, 4)$ intersects the line $x + y = 9$ at the point $P .$ The minimum distance of $A P,$ is