# 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

To find the intersecting points,

solve the system of equations:*x*^{2} + *y*^{2} = 25*y* = *x* + 1

Put *y* = *x* + 1

into *x*^{2} + *y*^{2} = 25.

Then *x*^{2} + (*x* + 1)^{2} = 25.

Substitution method

(*x* + 1)^{2} = *x*^{2} + 2⋅*x*⋅1 + 1^{2}

Square of a difference (*a* - *b*)^{2}

*x*^{2} + *x*^{2} = 2*x*^{2}

2⋅*x*⋅1 = 2*x*

1^{2} = 1

Move 25 to the left side.

Then 2*x*^{2} + 2*x* - 24 = 0.

Divide both sides by 2.

Then *x*^{2} + *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 *x*^{2} + *y*^{2} = 25

and the line *y* = *x* + 1

are (3, 4) and (-4, -3).