Intersection of a Line and a Curve

Question 1

To find the intersection of a line and a curve you should:

Accepted Answer

substitute the line equation into the curve and solve — Substituting the line into the curve gives one equation whose solutions are the intersection points.

Question 2

A line is a tangent to a curve when the discriminant of the resulting quadratic is:

Accepted Answer

equal to 0 — A repeated root (one point of contact) occurs when b² − 4ac = 0.

Question 3

The line y = x + 1 meets the curve y = x² at how many points?

Accepted Answer

2 — x² = x + 1 gives x² − x − 1 = 0; discriminant 1 + 4 = 5 > 0, so two points.

Question 4

The value of k for which y = x + k is a tangent to y = x² is:

Accepted Answer

−¼ — x² − x − k = 0 with discriminant 1 + 4k = 0 gives k = −¼.

Question 5

The line y = 2x − 3 meets y = x² − 2x + 1 at:

Accepted Answer

the point (2, 1) only — x² − 2x + 1 = 2x − 3 gives x² − 4x + 4 = (x − 2)² = 0, a tangent at (2, 1).

Question 6

The line y = mx meets y = x² + 1 at no points when:

Accepted Answer

−2 < m < 2 — x² − mx + 1 = 0 has no roots when m² − 4 < 0, i.e. −2 < m < 2.

Question 7

The points of intersection of y = x + 2 and y = x² are:

Accepted Answer

(2, 4) and (−1, 1) — x² = x + 2 gives (x − 2)(x + 1) = 0, so x = 2, −1 and points (2, 4), (−1, 1).

Question 8

If the discriminant of the substituted quadratic is greater than 0, the line and curve meet at:

Accepted Answer

two distinct points — A positive discriminant means two distinct real roots, hence two intersection points.

Question 9

How many points does the line y = 3x − 4 share with y = x²?

Accepted Answer

0 — x² − 3x + 4 = 0 has discriminant 9 − 16 = −7 < 0, so there are no intersection points.

Question 10

The line y = 1 meets the curve y = x² − 4x + 5 at:

Accepted Answer

the point (2, 1) only — x² − 4x + 5 = 1 gives (x − 2)² = 0, a tangent touching at (2, 1).

Substitution method

Tangency