14. Find the equation of the curve where slope at (x, y) is 9x and which passes through the origin (a) 9x2 = 2y (b) 9x 2 = 2y (c) 9x2 = -2y (d) -9x2 = 2y
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Answer:
9x
3
+9x
2
y−45x
2
=4y
3
+4xy
2
−20y
2
9x
2
(x+y−5)=4y
2
(x+y−5)
9x
2
(x+y−5)−4y
2
(x+y−5)=0
(9x
2
−4x
2
)(x+y−5)=0
(3x+2y)(3x−2x)(x+y−5)=0
so the three lines are
x=
3
2
y, x=
3
−2
y and x+y−5
from equation of lines , we get the vertices of triangle formed by the lines
A(0,0),B(−10,15) and C(2,3)
Area of the triangle =
2
1
(x
a
−x
c
)(y
b
−ya)−(x
a
−x
b
)(y
c
−y
a
)
=
2
1
(x
a
y
b
+x
b
y
c
+x
c
y
a
−x
a
y
c
−x
c
y
b
−x
b
y
a
)
=
2
1
(0×15+−10×3+2×0−0×3−2×15−(−10×0)
=
2
1
×−60
since, area is non- negativengle
So, the area of the triangle=30unit²
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