如图在反比例函数y=-2x和y=3x的图象上分别有A、B两点,若AB∥x轴且OA⊥OB,则OAOB=______.-数学

题目简介

如图在反比例函数y=-2x和y=3x的图象上分别有A、B两点,若AB∥x轴且OA⊥OB,则OAOB=______.-数学

题目详情

如图在反比例函数y=-
2
x
和y=
3
x
的图象上分别有A、B两点,若ABx轴且OA⊥OB,则
OA
OB
=______.
题型:解答题难度:中档来源:不详

答案

AB交y轴于C点,如图,
设B点坐标为(a,class="stub"3
a
),
∵ABx轴,
∴A点的纵坐标为class="stub"3
a
,OC⊥AB,
把y=class="stub"3
a
代入y=-class="stub"2
x
得x=-class="stub"2a
3
,则A点坐标为(-class="stub"2a
3
class="stub"3
a
),
∵OA⊥OB,
∴∠AOB=90°,
∴Rt△AOCRt△OBC,
class="stub"OA
OB
=class="stub"OC
BC
=class="stub"AC
OC
,即class="stub"OA
OB
=
class="stub"3
a
a
=
class="stub"2a
3
class="stub"3
a

class="stub"3
a
a
=
class="stub"2a
3
class="stub"3
a
得a4=class="stub"27
2

∴a2=
3
6
2

∴即class="stub"OA
OB
=
class="stub"3
a
a
class="stub"3
a2
=class="stub"3
3
6
2
=
6
3

故答案为
6
3

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