In the realm of Pure Mathematics, particularly within coordinate geometry, the concept of graph intersections is pivotal. This section aims to provide a comprehensive understanding of the relationship between graph intersection points and the solutions of equations. It explores the conditions under which a line intersects, touches, or does not meet a curve, employing both algebraic methods and graphical illustrations.

Intersection Points and Equation Solutions

• Intersection Points: These are the points where two or more graphs meet. They are crucial as they represent the solutions to the system of equations depicted by the graphs.
• Graphical Solution Method: To find solutions to a system of equations, plot each equation on the same set of axes. The coordinates of the intersection points are the solutions.

Conditions for Intersection

• Lines and Curves: A line can intersect a curve at no point, exactly one point (tangent), or multiple points.
• Determining Intersection Conditions: Algebraically solving the equations simultaneously helps determine the number of intersection points.

Lines Intersecting Curves

• Tangent Lines: A tangent line touches a curve at exactly one point without crossing it. The slope of the tangent at any point on a curve equals the derivative of the curve's equation at that point.
• Normal Lines: A normal line is perpendicular to a tangent at the point of tangency.

Algebraic Methods for Finding Intersections

Simultaneous Equations: Intersection points are found by solving the equations simultaneously.

Example:

Determine the intersection points of the line $y = x + 1$ and the circle $x^2 + y^2 = 4$.

Solution:

• Solving for $x$: The solutions are $x = -\frac{1}{2} + \frac{\sqrt{7}}{2}$ and $x = -\frac{1}{2} - \frac{\sqrt{7}}{2}$.
• Finding $y$-Coordinates:
  • For $x = -\frac{1}{2} + \frac{\sqrt{7}}{2}$, $y = \frac{1}{2} + \frac{\sqrt{7}}{2}$.
  • For $x = -\frac{1}{2} - \frac{\sqrt{7}}{2}$, $y = \frac{1}{2} - \frac{\sqrt{7}}{2}$.
• Intersection Points:
  • Point 1: $(-1/2 + \sqrt{7}/2, 1/2 + \sqrt{7}/2)$
  • Point 2: $(-1/2 - \sqrt{7}/2, 1/2 - \sqrt{7}/2)$

These points are where the line $y = x + 1$ and the circle $x^2 + y^2 = 4$intersect.

Practical Problems

Problem 1:

Find the intersection points of the line $y = 2x - 3$ and the circle $x^2 + y^2 = 25$.

Solution:

1. Combine Equations:

Substituting $y$ from the line equation into the circle equation results in $5x^2 - 12x - 16 = 0$.

2. Solve for $x$:

The quadratic equation is solved to find $x$ values:

• $x = \frac{6}{5} - \frac{2\sqrt{29}}{5}$ and $x = \frac{6}{5} + \frac{2\sqrt{29}}{5}$.

3. Find $y$ Values:

Corresponding $y$ values are calculated as:

• For $x = \frac{6}{5} - \frac{2\sqrt{29}}{5}$, $y = -\frac{4\sqrt{29}}{5} - \frac{3}{5}$.
• For $x = \frac{6}{5} + \frac{2\sqrt{29}}{5}$, $y = \frac{4\sqrt{29}}{5} - \frac{3}{5}$.

4. Intersection Points:

• Point 1: $\left(\frac{6}{5} - \frac{2\sqrt{29}}{5}, -\frac{4\sqrt{29}}{5} - \frac{3}{5}\right)$
• Point 2: $\left(\frac{6}{5} + \frac{2\sqrt{29}}{5}, \frac{4\sqrt{29}}{5} - \frac{3}{5}\right)$

These points are where the given line and circle intersect.

Problem 2:

Check if $y = -3x + 4$ is tangent to $x^2 + y^2 = 10$.

Solution:

1. Line Equation: $y = -3x + 4$

2. Circle Equation: $x^2 + y^2 = 10$

3. Substitute $y$ into Circle Equation:

$x^2 + (-3x + 4)^2 = 10$

$x^2 + 9x^2 - 24x + 16 = 10$

$10x^2 - 24x + 6 = 0$

4. Solve Quadratic Equation for $x$:

$a = 10, b = -24, c = 6$

$x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(10)(6)}}{2(10)}$

$x = \frac{24 \pm \sqrt{576 - 240}}{20}$

$x = \frac{24 \pm \sqrt{336}}{20}$

$x = \frac{24 \pm \sqrt{84}}{10}$

5. Check for Tangency: A line is tangent to a circle if the quadratic equation has exactly one solution, which occurs when the discriminant $(b^2 - 4ac)$ is zero.

Discriminant $= (-24)^2 - 4(10)(6) = 576 - 240 = 336$

Since the discriminant is not zero, the equation has two distinct solutions, indicating two intersection points between the line and the circle.

Conclusion: The line $y = -3x + 4$ is not tangent to the circle $x^2 + y^2 = 10$ as there are two intersection points.