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Byju's Answer
Standard XII
Mathematics
Position of a Point W.R.T Ellipse
Find the poin...
Question
Find the points on the curve y = x
3
− 2x
2
− 2x at which the tangent lines are parallel to the line y = 2x − 3.
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Solution
Let (x
1
, y
1
) be the required point.
Given:
y
=
2
x
-
3
∴
Slope
of
the
line
=
d
y
d
x
=
2
y
=
x
3
-
2
x
2
-
2
x
Since
x
1
y
1
lies
on
curve
,
y
1
=
x
1
3
-
2
x
1
2
-
2
x
1
.
.
.
1
⇒
d
y
d
x
x
1
,
y
1
=
3
x
1
2
-
4
x
1
-
2
It is given that the tangent and the given line are parallel.
∴ Slope of the tangent = Slope of the given line
3
x
1
2
-
4
x
1
-
2
=
2
⇒
3
x
1
2
-
4
x
1
-
4
=
0
⇒
3
x
1
2
-
6
x
1
+
2
x
1
-
4
=
0
⇒
3
x
1
x
1
-
2
+
2
x
1
-
2
=
0
⇒
x
1
-
2
3
x
1
+
2
=
0
⇒
x
1
=
2
or
x
1
=
-
2
3
Case
1
When
x
1
=
2
On
s
ubstituting the value of
x
1
in eq. (1), we get
y
1
=
8
-
8
-
4
=
-
4
∴
x
1
,
y
1
=
2
,
-
4
Case
2
When
x
1
=
-
2
3
On
s
ubstituting the value of
x
1
in eq. (1), we get
y
1
=
-
8
27
-
8
9
+
4
3
=
-
8
-
24
+
36
27
=
4
27
∴
x
1
,
y
1
=
-
2
3
,
4
27
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