Angle Between Curves at Intersection
On this page
Direct answer
Curves meeting at a point cut at the angle between their tangents there: for slopes m1 and m2, tan θ = |(m1 − m2)/(1 + m1 m2)|, and the intersection is orthogonal exactly when m1 m2 = −1. For implicit curves F(x, y) = 0, the slope comes from dy/dx = −Fx/Fy evaluated at the common point. Two degenerate readings carry the exam weight: if the denominator vanishes, the tangents are perpendicular (θ = 90°); if m1 = m2, the curves touch — tangency, angle zero. A classical bonus worth one statement: confocal ellipse and hyperbola intersect at right angles.
What you must remember
- The formula: tan θ = |(m1 − m2)/(1 + m1m2)|, the same one from straight lines, applied to the two tangent slopes at the intersection point.
- Orthogonality: m1 m2 = −1; in circle language the radius to the point is one of the tangents' normals.
- Tangency: m1 = m2 at a common point means the curves touch with zero angle — contact of order at least one.
- Implicit slope: for F(x, y) = 0, dy/dx = −(∂F/∂x)/(∂F/∂y) at the point — the circle x² + y² = r² has slope −x/y at (x, y).
- Parabola slope: y² = 4ax carries slope 2a/y at (x, y), or 1/t² at parameter t.
- Find every intersection first: missing a common point means missing a whole part of the answer — solve the pair completely before computing any angle.
- Confocal classic: every ellipse and hyperbola from the same pair of foci cut orthogonally, a result JEE Advanced has quoted directly.
Two parabolas meeting twice
Find the angles at which y = x² meets y = x³. Equating, x² = x³ gives x²(x − 1) = 0, so the common points are (0, 0) and (1, 1) — two intersections, two behaviours. At (1, 1) the slopes are 2 and 3, so tan θ = |(3 − 2)/(1 + 6)| = 1/7 and θ = tan⁻¹(1/7), a shallow cut. At (0, 0) both slopes equal 0: the tangents coincide and the curves touch, x³ running beneath x² throughout (0, 1) — an intersection without a cut. The lesson generalises: whenever the intersection equation has a repeated root, expect tangency at that point, and check the slope equality before declaring an angle. Now the orthogonal case in one line: the parabola y = x² + 1? A cleaner example is the pair y² = x and x² + y² = 2? They meet at (1, 1) and (1, −1); at (1, 1) the parabola's slope is 1/(2y) = 1/2 while the circle's is −1, giving product −1/2 — not orthogonal, and computing rather than guessing decided it.
How JEE frames it
JEE Main keeps the computation local: two curves, a named point, the angle from the formula, with the standard distractor being the complement (using cot θ values) or the acute-versus-obtuse twin from dropping the modulus. Implicit differentiation items dominate — the slope of the ellipse x²/25 + y²/9 = 1 at (3, 12/5)? comes from dy/dx = −9x/25y — and the angle question then rides on that slope. JEE Advanced goes structural: show that the curves x³ − 3xy² + 2 = 0? More realistically, prove orthogonality for a family, as with confocal conics, or find a member of a curve family cutting a given curve at a prescribed angle — the orthogonal-trajectories theme in disguise. The two habits that hold marks: first solve for all intersections (the y = x², y = x⁴ pair hides its tangency at the origin inside a repeated root), and second, evaluate slopes at the point, not anywhere else — substituting the point into the derivative is where silent sign errors enter.
Frequently asked questions
What is the angle between two curves at an intersection point?
The angle between their tangents at that point: tan θ = |(m1 − m2)/(1 + m1m2)| for the two slopes.
When do two curves intersect orthogonally?
When the product of their slopes at the common point is −1, each tangent perpendicular to the other.
How do you get the slope of an implicit curve?
Differentiate F(x, y) = 0 to get dy/dx = −Fx/Fy and evaluate at the point — no need to solve for y.
What does a repeated root in the intersection equation signal?
Tangency: the curves touch at that point with equal slopes and zero angle of intersection.
Which famous curve families cut at right angles?
Confocal ellipses and hyperbolas — sharing foci forces orthogonal intersection, the coordinate-geometry classic quoted directly in Advanced papers.