Trace the curve y² (a²+ x²) = x²(a²-x²), a>0
Answers
Answer:
The best way is to do it with the help of polar coordinates.
{
x
=
r
cos
θ
y
=
r
sin
θ
substituting
r
2
⋅
r
cos
θ
=
a
r
2
sin
2
θ
or for
r
>
0
r
=
a
sin
θ
tan
θ
we can compute easily some values for
r
given
θ
(
θ
0
π
4
π
2
r
0
√
2
infinity
)
We know also that the curve is symmetric regarding the
x
=
0
axis having also a vertical asymptote at
x
=
a
because
y
2
=
x
3
a
−
x
Attached a plot for
a
=
2
Answer:
The tangents to the curve at (a, 0) and (-a,0) are parallel to Y-axis.
The curve has no asymptotes.
Step-by-step explanation:
- A straight line is said to be an asymptote to an infinite branch of a curve if the perpendicular distance from a point on the curve to the given line approaches to zero as the point moves to infinity along the branch of curve.
- The asymptotes parallel to x-axis are called horizontal asymptotes, those which are parallel to y-axis are called vertical asymptotes and those which are neither parallel to x-axis nor parallel to y-axis are called oblique asymptotes.
- The equations of a vertical asymptotes are obtained by equating the coefficient of highest degree term in y to zero if it is not a constant.
- To obtain the equations of oblique asymptotes, substitutes y = mx + c in the given equation. then equate the coefficients of the highest degree term in x and next highest degree term in x to zero, if it is not a constant, to determine m and c. If the values of m and c exists, then y = mx + c is the equation of the oblique asymptote.
(i) Symmetry: Symmetrical about both the axes.
(ii) Points: Passes through (0, 0), (a, 0) and (-a,0). Loop between (0, 0) and (a, 0); also between (0, 0) and (-a, 0).
(iii) Tangents: Lowest degree terms a2y2-x2a2 = 0, we get y = ± x, the tangents.
(iv) Region:
Note that y is imaginary if |x|>a. Hence the whole curve lies between x = a and x = -a.
Therefore the tangents to the curve at (a, 0) and (-a,0) are parallel to Y-axis.
The curve has no asymptotes.
Reference Link
- https://brainly.in/question/12435862
- https://brainly.in/question/314520
