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Page 1
JEE Solved Example on Ellipse
JEE Mains
Q1: The equation
?? ?? ?? -?? +
?? ?? ?? -?? + ?? = ?? represents an ellipse, if
(A) ?? > ??
(B) ?? < ?? < ??
(C) ?? > ??
(D) ?? ? {?? , ?? }
Ans: (B)
?? ?? - 2
+
?? 2
5 - ?? = 1
? ?? - 2 > 0 and 5 - ?? > 0
? ?? ? ( 2,5)
Q2: The eccentricity of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? is
(A)
?? ??
(B)
?? ??
(C)
v ?? ??
(D)
v ?? ??
Ans: (D) 4?? 2
+ 8?? + 9?? 2
+ 36?? + 4 = 0
? ( 2?? + 2)
2
+ ( 3?? + 6)
2
= 36
?
( ?? + 1)
2
9
+
( ?? + 2)
2
4
= 1
? ?? =
v
1 -
4
9
=
v 5
3
Q3: If ?????? ?? ?? · ?????? ?? ?? = -
?? ?? ?? ?? then the chord joining two points ?? ?? and ?? ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =
?? will subtend a right angle at:
(A) Focus
(B) Centre
(C) End of the major axis
(D) End of the minor axis
Page 2
JEE Solved Example on Ellipse
JEE Mains
Q1: The equation
?? ?? ?? -?? +
?? ?? ?? -?? + ?? = ?? represents an ellipse, if
(A) ?? > ??
(B) ?? < ?? < ??
(C) ?? > ??
(D) ?? ? {?? , ?? }
Ans: (B)
?? ?? - 2
+
?? 2
5 - ?? = 1
? ?? - 2 > 0 and 5 - ?? > 0
? ?? ? ( 2,5)
Q2: The eccentricity of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? is
(A)
?? ??
(B)
?? ??
(C)
v ?? ??
(D)
v ?? ??
Ans: (D) 4?? 2
+ 8?? + 9?? 2
+ 36?? + 4 = 0
? ( 2?? + 2)
2
+ ( 3?? + 6)
2
= 36
?
( ?? + 1)
2
9
+
( ?? + 2)
2
4
= 1
? ?? =
v
1 -
4
9
=
v 5
3
Q3: If ?????? ?? ?? · ?????? ?? ?? = -
?? ?? ?? ?? then the chord joining two points ?? ?? and ?? ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =
?? will subtend a right angle at:
(A) Focus
(B) Centre
(C) End of the major axis
(D) End of the minor axis
Ans: (B) Let P be ( ?? 1
) and Q = ( ?? 2
)
Slope ???? =
?? ?? tan ?? 1
and slope ???? =
?? ?? tan ?? 2
? ?? ????
× ?? ????
=
?? 2
?? 2
× (
-?? 2
?? 2
) = -1
? It subtends 90
°
at centre
Q4: If the line ?? = ?? ?? + ?? be a tangent to the ellipse
?? ?? ?? +
?? ?? ?? = ?? , then ?? is equal to
(A) ±??
(B) ±??
(C) ±??
(D) ±??
Ans: (B) ?? 2
= ?? 2
?? 2
+ ?? 2
? ?? 2
= 8 × 4 + 4
? ?? = ±v 36 = ±6
Q5: If the line ?? ?? + ?? ?? = -v ?? touches the ellipse ?? ?? ?? + ?? ?? ?? = ?? then, the point of contact is
(A) (
?? v ?? ,
?? v ?? )
(B) (
?? v ?? ,
-?? v ?? )
(C) (
?? v ?? ,
-?? v ?? )
(D) (
-?? v ?? ,
-?? v ?? )
Ans: (D) Let ( ?? 1
, ?? 1
) be point of contact to ellipse ? 3xx
1
+ 4yy
1
= 1 is equation of tangent at (
x
1
, y
1
)
?
3?? 1
3
=
4?? 1
4
= -
1
v 7
? ( ?? 1
, ?? 1
)= ( -
1
v 7
, -
1
v 7
)
Page 3
JEE Solved Example on Ellipse
JEE Mains
Q1: The equation
?? ?? ?? -?? +
?? ?? ?? -?? + ?? = ?? represents an ellipse, if
(A) ?? > ??
(B) ?? < ?? < ??
(C) ?? > ??
(D) ?? ? {?? , ?? }
Ans: (B)
?? ?? - 2
+
?? 2
5 - ?? = 1
? ?? - 2 > 0 and 5 - ?? > 0
? ?? ? ( 2,5)
Q2: The eccentricity of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? is
(A)
?? ??
(B)
?? ??
(C)
v ?? ??
(D)
v ?? ??
Ans: (D) 4?? 2
+ 8?? + 9?? 2
+ 36?? + 4 = 0
? ( 2?? + 2)
2
+ ( 3?? + 6)
2
= 36
?
( ?? + 1)
2
9
+
( ?? + 2)
2
4
= 1
? ?? =
v
1 -
4
9
=
v 5
3
Q3: If ?????? ?? ?? · ?????? ?? ?? = -
?? ?? ?? ?? then the chord joining two points ?? ?? and ?? ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =
?? will subtend a right angle at:
(A) Focus
(B) Centre
(C) End of the major axis
(D) End of the minor axis
Ans: (B) Let P be ( ?? 1
) and Q = ( ?? 2
)
Slope ???? =
?? ?? tan ?? 1
and slope ???? =
?? ?? tan ?? 2
? ?? ????
× ?? ????
=
?? 2
?? 2
× (
-?? 2
?? 2
) = -1
? It subtends 90
°
at centre
Q4: If the line ?? = ?? ?? + ?? be a tangent to the ellipse
?? ?? ?? +
?? ?? ?? = ?? , then ?? is equal to
(A) ±??
(B) ±??
(C) ±??
(D) ±??
Ans: (B) ?? 2
= ?? 2
?? 2
+ ?? 2
? ?? 2
= 8 × 4 + 4
? ?? = ±v 36 = ±6
Q5: If the line ?? ?? + ?? ?? = -v ?? touches the ellipse ?? ?? ?? + ?? ?? ?? = ?? then, the point of contact is
(A) (
?? v ?? ,
?? v ?? )
(B) (
?? v ?? ,
-?? v ?? )
(C) (
?? v ?? ,
-?? v ?? )
(D) (
-?? v ?? ,
-?? v ?? )
Ans: (D) Let ( ?? 1
, ?? 1
) be point of contact to ellipse ? 3xx
1
+ 4yy
1
= 1 is equation of tangent at (
x
1
, y
1
)
?
3?? 1
3
=
4?? 1
4
= -
1
v 7
? ( ?? 1
, ?? 1
)= ( -
1
v 7
, -
1
v 7
)
Q6: The point of intersection of the tangents at the point ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? and its
corresponding point ?? on the auxiliary circle meet on the line:
(A) ?? =
?? ??
(B) ?? = ??
(C) ?? = ??
(D) None of these
Ans: (C) x-axis
Q7: The equation of the normal to the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? at the positive and of latus rectum
(A) ?? + ???? + ?? ?? ?? = ??
(B) ?? - ???? - ?? ?? ?? = ??
(C) ?? - ???? - ?? ?? ?? = ??
(D) None of these
Ans: (B) The positive end of latus rectum is ( ?? ?? 2
?? 2
?? )
acos ?? = ae ? cos ?? = e
bsin ?? =
b
2
a
2
&sin ?? =
b
a
equation of normal is
????
cos ?? -
????
sin ?? = ?? 2
- ?? 2
?
ax
e
-
by
b
× a = a
2
( e
2
)
or ?? - ???? = ?? ?? 3
? ?? - ???? - ?? 3
?? = 0
Q8: The normal at an end of a latus rectum of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? passes through an end of the
minor axis, if:
(A) ?? ?? + ?? ?? = ??
(B) ?? ?? + ?? ?? = ??
(C) ?? ?? + ?? = ??
(D) ?? ?? + ?? = ??
Page 4
JEE Solved Example on Ellipse
JEE Mains
Q1: The equation
?? ?? ?? -?? +
?? ?? ?? -?? + ?? = ?? represents an ellipse, if
(A) ?? > ??
(B) ?? < ?? < ??
(C) ?? > ??
(D) ?? ? {?? , ?? }
Ans: (B)
?? ?? - 2
+
?? 2
5 - ?? = 1
? ?? - 2 > 0 and 5 - ?? > 0
? ?? ? ( 2,5)
Q2: The eccentricity of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? is
(A)
?? ??
(B)
?? ??
(C)
v ?? ??
(D)
v ?? ??
Ans: (D) 4?? 2
+ 8?? + 9?? 2
+ 36?? + 4 = 0
? ( 2?? + 2)
2
+ ( 3?? + 6)
2
= 36
?
( ?? + 1)
2
9
+
( ?? + 2)
2
4
= 1
? ?? =
v
1 -
4
9
=
v 5
3
Q3: If ?????? ?? ?? · ?????? ?? ?? = -
?? ?? ?? ?? then the chord joining two points ?? ?? and ?? ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =
?? will subtend a right angle at:
(A) Focus
(B) Centre
(C) End of the major axis
(D) End of the minor axis
Ans: (B) Let P be ( ?? 1
) and Q = ( ?? 2
)
Slope ???? =
?? ?? tan ?? 1
and slope ???? =
?? ?? tan ?? 2
? ?? ????
× ?? ????
=
?? 2
?? 2
× (
-?? 2
?? 2
) = -1
? It subtends 90
°
at centre
Q4: If the line ?? = ?? ?? + ?? be a tangent to the ellipse
?? ?? ?? +
?? ?? ?? = ?? , then ?? is equal to
(A) ±??
(B) ±??
(C) ±??
(D) ±??
Ans: (B) ?? 2
= ?? 2
?? 2
+ ?? 2
? ?? 2
= 8 × 4 + 4
? ?? = ±v 36 = ±6
Q5: If the line ?? ?? + ?? ?? = -v ?? touches the ellipse ?? ?? ?? + ?? ?? ?? = ?? then, the point of contact is
(A) (
?? v ?? ,
?? v ?? )
(B) (
?? v ?? ,
-?? v ?? )
(C) (
?? v ?? ,
-?? v ?? )
(D) (
-?? v ?? ,
-?? v ?? )
Ans: (D) Let ( ?? 1
, ?? 1
) be point of contact to ellipse ? 3xx
1
+ 4yy
1
= 1 is equation of tangent at (
x
1
, y
1
)
?
3?? 1
3
=
4?? 1
4
= -
1
v 7
? ( ?? 1
, ?? 1
)= ( -
1
v 7
, -
1
v 7
)
Q6: The point of intersection of the tangents at the point ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? and its
corresponding point ?? on the auxiliary circle meet on the line:
(A) ?? =
?? ??
(B) ?? = ??
(C) ?? = ??
(D) None of these
Ans: (C) x-axis
Q7: The equation of the normal to the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? at the positive and of latus rectum
(A) ?? + ???? + ?? ?? ?? = ??
(B) ?? - ???? - ?? ?? ?? = ??
(C) ?? - ???? - ?? ?? ?? = ??
(D) None of these
Ans: (B) The positive end of latus rectum is ( ?? ?? 2
?? 2
?? )
acos ?? = ae ? cos ?? = e
bsin ?? =
b
2
a
2
&sin ?? =
b
a
equation of normal is
????
cos ?? -
????
sin ?? = ?? 2
- ?? 2
?
ax
e
-
by
b
× a = a
2
( e
2
)
or ?? - ???? = ?? ?? 3
? ?? - ???? - ?? 3
?? = 0
Q8: The normal at an end of a latus rectum of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? passes through an end of the
minor axis, if:
(A) ?? ?? + ?? ?? = ??
(B) ?? ?? + ?? ?? = ??
(C) ?? ?? + ?? = ??
(D) ?? ?? + ?? = ??
Ans: (A) Consider normal at positive end of latus rectum from above equation of normal is ?? - ???? -
?? 3
?? = 0
It passes through ( 0, -?? )
? ???? - ?? 3
?? = 0
? ?? - ?? 2
?? = 0
?
b
a
= e
2
? 1 - e
2
= e
4
or e
2
+ e
4
= 1
Q9: If ???? is perpendicular from the centre of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? to the tangent at ?? and ?? is
the point where the normal at ?? meets the major axis, then the product CF.PG is:
(A) ?? ??
(B) ?? ?? ??
(C) ?? ??
(D) ?? ?? - ?? ??
Ans: (C) Tangent at ?? is
?? cos ?? ?? +
?? sin ?? ?? - 7 = 0
? ???? =
1
v
cos
2
?? ?? 2
+
sin
2
?? ?? 2
?? = ( ?? cos ?? , ?? sin ?? )
Equation of normal at
?? =
????
cos ?? -
????
sin ?? = ?? 2
- ?? 2
? ?? = (
( ?? 2
- ?? 2
) cos ?? ?? , 0)
? ???? =
v
?? 2
sin
2
?? +
?? 4
?? 2
cos
2
?? = ?? 2
v
sin
2
?? ?? 2
+
cos
2
?? ?? 2
? ???? . ???? = ?? 2
Page 5
JEE Solved Example on Ellipse
JEE Mains
Q1: The equation
?? ?? ?? -?? +
?? ?? ?? -?? + ?? = ?? represents an ellipse, if
(A) ?? > ??
(B) ?? < ?? < ??
(C) ?? > ??
(D) ?? ? {?? , ?? }
Ans: (B)
?? ?? - 2
+
?? 2
5 - ?? = 1
? ?? - 2 > 0 and 5 - ?? > 0
? ?? ? ( 2,5)
Q2: The eccentricity of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? is
(A)
?? ??
(B)
?? ??
(C)
v ?? ??
(D)
v ?? ??
Ans: (D) 4?? 2
+ 8?? + 9?? 2
+ 36?? + 4 = 0
? ( 2?? + 2)
2
+ ( 3?? + 6)
2
= 36
?
( ?? + 1)
2
9
+
( ?? + 2)
2
4
= 1
? ?? =
v
1 -
4
9
=
v 5
3
Q3: If ?????? ?? ?? · ?????? ?? ?? = -
?? ?? ?? ?? then the chord joining two points ?? ?? and ?? ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =
?? will subtend a right angle at:
(A) Focus
(B) Centre
(C) End of the major axis
(D) End of the minor axis
Ans: (B) Let P be ( ?? 1
) and Q = ( ?? 2
)
Slope ???? =
?? ?? tan ?? 1
and slope ???? =
?? ?? tan ?? 2
? ?? ????
× ?? ????
=
?? 2
?? 2
× (
-?? 2
?? 2
) = -1
? It subtends 90
°
at centre
Q4: If the line ?? = ?? ?? + ?? be a tangent to the ellipse
?? ?? ?? +
?? ?? ?? = ?? , then ?? is equal to
(A) ±??
(B) ±??
(C) ±??
(D) ±??
Ans: (B) ?? 2
= ?? 2
?? 2
+ ?? 2
? ?? 2
= 8 × 4 + 4
? ?? = ±v 36 = ±6
Q5: If the line ?? ?? + ?? ?? = -v ?? touches the ellipse ?? ?? ?? + ?? ?? ?? = ?? then, the point of contact is
(A) (
?? v ?? ,
?? v ?? )
(B) (
?? v ?? ,
-?? v ?? )
(C) (
?? v ?? ,
-?? v ?? )
(D) (
-?? v ?? ,
-?? v ?? )
Ans: (D) Let ( ?? 1
, ?? 1
) be point of contact to ellipse ? 3xx
1
+ 4yy
1
= 1 is equation of tangent at (
x
1
, y
1
)
?
3?? 1
3
=
4?? 1
4
= -
1
v 7
? ( ?? 1
, ?? 1
)= ( -
1
v 7
, -
1
v 7
)
Q6: The point of intersection of the tangents at the point ?? on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? and its
corresponding point ?? on the auxiliary circle meet on the line:
(A) ?? =
?? ??
(B) ?? = ??
(C) ?? = ??
(D) None of these
Ans: (C) x-axis
Q7: The equation of the normal to the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? at the positive and of latus rectum
(A) ?? + ???? + ?? ?? ?? = ??
(B) ?? - ???? - ?? ?? ?? = ??
(C) ?? - ???? - ?? ?? ?? = ??
(D) None of these
Ans: (B) The positive end of latus rectum is ( ?? ?? 2
?? 2
?? )
acos ?? = ae ? cos ?? = e
bsin ?? =
b
2
a
2
&sin ?? =
b
a
equation of normal is
????
cos ?? -
????
sin ?? = ?? 2
- ?? 2
?
ax
e
-
by
b
× a = a
2
( e
2
)
or ?? - ???? = ?? ?? 3
? ?? - ???? - ?? 3
?? = 0
Q8: The normal at an end of a latus rectum of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? passes through an end of the
minor axis, if:
(A) ?? ?? + ?? ?? = ??
(B) ?? ?? + ?? ?? = ??
(C) ?? ?? + ?? = ??
(D) ?? ?? + ?? = ??
Ans: (A) Consider normal at positive end of latus rectum from above equation of normal is ?? - ???? -
?? 3
?? = 0
It passes through ( 0, -?? )
? ???? - ?? 3
?? = 0
? ?? - ?? 2
?? = 0
?
b
a
= e
2
? 1 - e
2
= e
4
or e
2
+ e
4
= 1
Q9: If ???? is perpendicular from the centre of the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? = ?? to the tangent at ?? and ?? is
the point where the normal at ?? meets the major axis, then the product CF.PG is:
(A) ?? ??
(B) ?? ?? ??
(C) ?? ??
(D) ?? ?? - ?? ??
Ans: (C) Tangent at ?? is
?? cos ?? ?? +
?? sin ?? ?? - 7 = 0
? ???? =
1
v
cos
2
?? ?? 2
+
sin
2
?? ?? 2
?? = ( ?? cos ?? , ?? sin ?? )
Equation of normal at
?? =
????
cos ?? -
????
sin ?? = ?? 2
- ?? 2
? ?? = (
( ?? 2
- ?? 2
) cos ?? ?? , 0)
? ???? =
v
?? 2
sin
2
?? +
?? 4
?? 2
cos
2
?? = ?? 2
v
sin
2
?? ?? 2
+
cos
2
?? ?? 2
? ???? . ???? = ?? 2
Q???? : ?? - ?? ?? + ?? = ?? is a common tangent to ?? ?? = ?? ?? and
?? ?? ?? +
?? ?? ?? ?? = ?? . Then the value of ?? and
the other common tangent are given by:
(A) ?? = v ?? ; ?? + ?? ?? + ?? = ??
(B) ?? = ?? ; ?? + ?? ?? + ?? = ??
(C) ?? = v ?? ; ?? + ?? ?? - ?? = ??
(D) ?? = v ?? ; ?? - ?? ?? - ?? = ??
Ans: (A) ?? =
?? 2
+ 2
? ?? 2
= ?? 2
?? 2
+ ?? 2
? 4 = 4 ×
1
4
+ ?? 2
? ?? = v 3
? The other common tangent has slope -m
? ?? =
1
-
1
2
= -2
? Equation is ?? = -
1
2
?? - 2
or ?? + 2?? + 2 = 0
Q11: An ellipse is such that the length of the latus rectum is equal to the sum of the lengths of its
semi principal axes. Then:
(A) Ellipse bulges to a circle
(B) Ellipse becomes a line segment between the two foci
(C) Ellipse becomes a parabola
(D) None of these
Ans: (A)
2?? 2
?? = ?? + ??
or 2?? 2
= ?? 2
+ ????
?? 2
+ ???? - 2?? 2
= 0
?? = -
?? ± v ?? 2
+ 8?? 2
2
?? = -
?? + 3?? 2
= ??
? Ellipse bulges to circle
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