Ex 2: Find a Point of Intersection of a Line and a Circle

The video introduces the objective: finding the intersection point of line y = 2x + 1 and a circle with center (0, 4) and radius 4 in the first quadrant. The equation for the circle is derived as x^2 + (y-4)^2 = 16.

Using the substitution method, the linear equation is substituted into the circle equation. The expression is expanded and rearranged to form a quadratic equation: 5x^2 - 12x - 7 = 0.

Since the quadratic equation is not easily factorable, the quadratic formula is applied with coefficients a=5, b=-12, c=-7. Only the positive root is kept to satisfy the requirement of being in the first quadrant.

The X-coordinate is calculated and rounded to 2.885. This value is then back-substituted into the linear equation y = 2x + 1 to solve for the Y-coordinate, resulting in 6.770.