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(#M40027021) KVPY question Mental Ability Keep an EYE

Let A = (4, 0), B = (0, 12) be two points in the plane. The locus of a point C such that the area of triangle
ABC is 18 sq. units is

Asked In KVPY MAN (12 years ago)

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pooja  (6 years ago)

The locus of point C will be 2 parallel lines
L
1
and
L
2
.
This is because the distance between
L
1
and
L
or
L
2
and
L
is equal to h.
Thus area formed by corresponding triangle will have area
=
18
sq. units.
Equation of line
L
:
3
x
+
y
−
12
=
0
Let equations of line
L
1
and
L
2
be defined by parameter
c
:
3
x
+
y
+
c
=
0
.... eqn(i)
d
=
√
(
4
−
0
)
2
+
(
12
−
0
)
2
=
4
√
10
units
Area
=
1
2
×
d
×
h
=
18
⇒
h
=
18
×
2
4
√
10
=
9
√
10
Using distance formula for lines:
|
c
−
(
−
12
)
|
√
(
3
2
+
1
2
)
=
h
=
9
√
1
0
⇒
|
c
+
12
|
=
9
⇒
c
=
9
−
12
=
−
3
OR
c
=
−
9
−
12
=
−
21
Substituting back in eqn(i) gives,
⇒
y
+
3
x
−
12
=
−
9
OR
y
+
3
x
−
12
=
9
⇒

(
y
+
3
x
−
12
)
2
=
81

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