Intersecting Points of a Circle and a Secant

Intersecting Points of a Circle and a Secant

How to find the coordinates of the intersecting points of a circle and a secant: example and its solution.

Example

For the given equations below, find the coordinates of the intersecting points. x^2 + y^2 = 25, y = x + 1.

To find the intersecting points,
solve the system of equations:

x2 + y2 = 25
y = x + 1

Put y = x + 1
into x2 + y2 = 25.

Then x2 + (x + 1)2 = 25.

Substitution method

(x + 1)2 = x2 + 2⋅x⋅1 + 12

Square of a difference (a - b)2

x2 + x2 = 2x2
2⋅x⋅1 = 2x
12 = 1

Move 25 to the left side.

Then 2x2 + 2x - 24 = 0.

Divide both sides by 2.

Then x2 + x - 12 = 0.

Factor the left side.

Factor a quadratic trinomial

Find a pair of numbers
whose product is the constant term [-12]
and whose sum is the middle term's coefficient [+1].

The constant term is (-).
So the signs of the numbers are different:
one is (+), and the other is (-).

(-1, 12) and (-2, 6)
are not the right numbers.

[-12] = -3⋅4
-3 + 4 = 1 = [+1]
So -3 and 4 are the right numbers.

Use -3 and +4
to write a factored form:
(x - 3)(x + 4) = 0.

Solve (x - 3)(x + 4) = 0.

Solving a quadratic equation by factoring

1) x - 3 = 0
So x = 3.

To find the y value of the intersecting point,
put x = 3
into y = x + 1.

Then y = 4.

So the intersecting point for case 1 is
(3, 4).

2) x + 4 = 0
So x = -4.

Put x = -4
into y = x + 1.

Then y = 3.

So the intersecting point for case 2 is
(-4, -3).

So the intersecting points are
(3, 4) and (-4, -3).

Let's see what you've solved.

The intersecting points of
the circle x2 + y2 = 25
and the line y = x + 1
are (3, 4) and (-4, -3).