Match the following lists:a. The coordinates of a point on the line x $$=$$ $$4y$$ $$1$$ $$5$$, z $$=$$ $$3y$$ $$-$$ $$6$$ at a distance $$3$$ from the point (5$$,$$3$$,$$-6) is $$/$$ areb. The plane containing the lines x−$$23=y+35=z+57$$ and parallel to $$i^+4j^+7k^$$ has the pointc. A line passes through two points $$A(2$$,$$-$$ $$3$$, $$-1)$$ and $$B(8$$, $$-1$$,$$2)$$. The coordinates of a point on this line nearer to the origin and at a distance of $$14$$ units from A $$is/ared$$. The coordinates of the foot of the perpendicular from the point (3$$, $$-1$$, $$1$$,$$1) on the line $$x2=y$$−$$23=z$$−$$34$$ is $$/$$ are

Question

Match the following lists:a. The coordinates of a point on the line x $$=$$ $$4y$$ $$1$$ $$5$$, z $$=$$ $$3y$$ $$-$$ $$6$$ at a distance $$3$$ from     the point (5$$,$$3$$,$$-6) is $$/$$ areb. The plane containing the lines x−$$23=y+35=z+57$$  and      parallel to $$i^+4j^+7k^$$   has the pointc. A line passes through two points $$A(2$$,$$-$$ $$3$$, $$-1)$$ and $$B(8$$, $$-1$$,$$2)$$. The coordinates of     a point on this line  nearer to the origin and at a distance of $$14$$ units from A $$is/ared$$. The coordinates of the foot of the perpendicular from the point (3$$, $$-1$$, $$1$$,$$1) on      the line $$x2=y$$−$$23=z$$−$$34$$ is $$/$$ are

Infinity Learn · Accepted Answer

a. The given line is $$x=$$ $$4y$$ $$+$$ $$5$$, $$z=3y$$ $$-$$ $$6$$    or  x−$$54=y$$,$$z+63=yor$$  x−$$54=y1=z+63=$$λ     Any point on the line is of the form  (4$$λ$$+5$$,λ,$$3$$λ$$-6)     The distance between  (4$$λ$$+5$$,λ,$$3$$λ$$-6) and (5$$, $$3$$, $$-$$ $$6) is $$3$$ units (given). Therefore      (4$$λ$$+5$$−$$5)2+($$λ−$$3)2+(3$$λ−$$6+6)2=9or$$  $$16$$λ$$2+$$λ$$2+9$$−$$6$$λ$$+9$$λ$$2=9or$$  $$26$$λ$$2$$−$$6$$λ$$=0or$$ λ$$=0$$,$$3/13$$      The point is $$(5$$, $$0$$, $$-$$ $$6).b.  The equation of the plane containing the lines  x−$$23=y+35=z+57$$ and parallel to $$i^+4j^+7k^$$ is       x−$$2y+3z+5147357=0or$$  x−$$2y+z$$−$$3=0$$     Point $$(-1$$,$$-2$$,$$0)$$ lies on the plane c.  The line passing through points $$A(2$$,$$-3$$,$$-$$ $$1)$$ and $$B(8$$,$$-1$$,$$2)is$$  x−$$28$$−$$2=y+3$$−$$1+3=z+12+1$$      or  x−$$26=y+32=z+13=$$λ       Any point on this line is of the form  $$P(6$$λ$$+22$$λ−$$3$$,$$3$$λ−$$1)$$ whose distance from point $$A(2$$, $$-$$ $$3$$, $$-$$ $$1)$$ is $$14$$ units.        Therefore,         $$PA=14PA2=(14)2$$⇒(6$$λ$$)2+(2$$λ$$)2+(3$$λ$$)2=196or$$ $$49$$λ$$2=196or$$ λ$$2=4or$$  λ$$=$$±$$2d$$.    Any point on line AB $$x2=y$$−$$23=z$$−$$34=$$λ is  $$M(2$$λ,$$3$$λ$$+2$$,$$4$$λ$$+3)         Therefore, the direction ratios of PM are $$2$$λ−$$3$$,$$3$$λ$$+3$$ and $$4$$λ−$$8$$          but  PM⊥AB∴$$2(2$$λ−$$3)+3(3$$λ$$+3)+4(4$$λ−$$8)=04$$λ−$$6+9$$λ$$+9+16$$λ−$$32=029$$λ−$$29=0$$λ$$=1$$         Therefore, foot of the perpendicular is $$M(2$$,$$5$$,$$7)$$.

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App