SME
Calculus I · Problem type
Implicit slope on a circle
Use this for “Find dy/dx on x² + y² = 100 at (6, −8), and the tangent line there.”
Diagram
x² + y² = 25
Center (0, 0)
radius 5.
The point (3, 4) is on the curve.
Tangent
at
(3,
4),
square
to
the
radius:
slope
−
x₀
y₀
=
−0.75;
y
=
−0.75x
+
6.25
This is an example. Type your own values to start a new problem.
x²
+
y²
=
r²
dy
dx
at
(
x₀
,
y₀
)
=
dy/dx
tangent:
y
=
dy/dx
x
+
c
r
Radius
worked out (not typed)
Formulas
r
2
=
x₀
2
+
y₀
2
25
=
3
2
+
4
2
r
=
√
r
2
5
=
√
25
dy/dx
=
−
x₀
y₀
−0.75
=
−
3
4
c
=
y₀
−
(dy/dx)
×
x₀
6.25 = 4 − (−0.75) × 3
Assumptions
•
x² + y² = r² is a circle with its center at the origin. Near a point on it, y is a function of x (the top or bottom half).
•
Differentiate both sides in x, using the chain rule on y²: 2x + 2y·(dy/dx) = 0, so dy/dx = −x ÷ y.
•
The radius to the point has slope y₀ ÷ x₀ (for x₀ ≠ 0), and the tangent −x₀ ÷ y₀: their product is −1, so the tangent is at right angles to the radius.
•
Where y₀ = 0 (the far left and right of the circle) the tangent is the vertical line x = x₀, which has no slope.
Step-by-step
WE KNOW
x₀ = 3, y₀ = 4
FIND
right side x² + y² (r²), radius (r), slope of the tangent (dy/dx), tangent’s y-intercept (c)
Step 1 · Find right side x² + y² (r²)
Use r² = x₀² + y₀²
The
point
is
on
the
circle,
so
it
makes
x
2
+
y
2
true.
r
2
=
3
2
+
4
2
r
2
=
9
+
16
r
2
=
25
Step 2 · Find radius (r)
Use r = √r²
The radius is the square root of the right side.
r
=
√
25
r
=
5
Step 3 · Find slope of the tangent (dy/dx)
Use dy/dx = −x₀ ÷ y₀
Solve 2x + 2y·(dy/dx) = 0 for dy/dx, then put in the point.
dy/dx
=
−
3
4
dy/dx = −0.75
Step 4 · Find tangent’s y-intercept (c)
Use c = y₀ − (dy/dx) × x₀
The
tangent
y
=
(dy/dx)·x
+
c
goes
through
the
point,
so
y₀
=
(dy/dx)·
x₀
+
c
.
Take
(dy/dx)·
x₀
from
y₀
.
c
=
4
−
(−0.75)
×
3
c
=
4
−
(−2.25)
c
=
6.25
Check
Put the numbers back in. Both sides should match.
25
=
3
2
+
4
2
✓
5
=
√
25
✓
−0.75
=
−
3
4
✓
6.25 = 4 − (−0.75) × 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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