If x=root3+root2/root3-root 2 &y=1/x,then find the value of x^2+xy-y^2
Answers
Step-by-step explanation:
x=√3+√2/√3-√2
rationalise denominator
we have,
x=5+2√6
y=1/5+2√6
again rationalise denominator
y=5-2√6
x^2+xy-y^2
=(5+2√6)^2+(5+2√6)(5-2√6)-(5-2√6)^2
=(49+20√6)+(1)-(49-2√6)
=49+20√6+1-49+20√6
=40√6+1
Answer:
Answer:
Value of x² + y² + xy is 99.
Step-by-step explanation:
Given:
x=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\:\:and\:\:y=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}x=
3
+
2
3
−
2
andy=
3
−
2
3
+
2
To find: value of x² + y² + xy
First we find,
xy=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}xy=
3
+
2
3
−
2
×
3
−
2
3
+
2
=\frac{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=
(
3
+
2
)(
3
−
2
)
(
3
−
2
)(
3
+
2
)
=\frac{(\sqrt{3})^2-(\sqrt{2})^2}{(\sqrt{3})^2-(\sqrt{2})^2}=
(
3
)
2
−(
2
)
2
(
3
)
2
−(
2
)
2
=\frac{3-2}{3-2}=
3−2
3−2
= 1
x^2=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\times\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}x
2
=
3
+
2
3
−
2
×
3
+
2
3
−
2
=\frac{(\sqrt{3}-\sqrt{2})^2}{(\sqrt{3}+\sqrt{2})^2}=
(
3
+
2
)
2
(
3
−
2
)
2
=\frac{3+2-2\sqrt{3}\sqrt{2}}{3+2+2\sqrt{3}\sqrt{2}}=
3+2+2
3
2
3+2−2
3
2
=\frac{5-2\sqrt{6}}{5+2\sqrt{6}}=
5+2
6
5−2
6
=\frac{5-2\sqrt{6}}{5+2\sqrt{6}}\times\frac{5-2\sqrt{6}}{5-2\sqrt{6}}=
5+2
6
5−2
6
×
5−2
6
5−2
6
=\frac{(5-2\sqrt{6})^2}{(5+2\sqrt{6})(5-2\sqrt{6})}=
(5+2
6
)(5−2
6
)
(5−2
6
)
2
=\frac{25+24-20\sqrt{6}}{25-24}=
25−24
25+24−20
6
= 49 - 20√6
y^2=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\times\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}y
2
=
3
−
2
3
+
2
×
3
−
2
3
+
2
=\frac{(\sqrt{3}+\sqrt{2})^2}{(\sqrt{3}-\sqrt{2})^2}=
(
3
−
2
)
2
(
3
+
2
)
2
=\frac{3+2+2\sqrt{3}\sqrt{2}}{3+2-2\sqrt{3}\sqrt{2}}=
3+2−2
3
2
3+2+2
3
2
=\frac{5+2\sqrt{6}}{5-2\sqrt{6}}=
5−2
6
5+2
6
=\frac{5+2\sqrt{6}}{5-2\sqrt{6}}\times\frac{5+2\sqrt{6}}{5+2\sqrt{6}}=
5−2
6
5+2
6
×
5+2
6
5+2
6
=\frac{(5+2\sqrt{6})^2}{(5+2\sqrt{6})(5-2\sqrt{6})}=
(5+2
6
)(5−2
6
)
(5+2
6
)
2
=\frac{25+24+20\sqrt{6}}{25-24}=
25−24
25+24+20
6
= 49 + 20√6
Now,
x² + y² + xy = 49 - 20√6 + 49 + 20√6 + 1 = 49 + 49 + 1 = 99
Therefore, Value of x² + y² + xy is 99