4. In many interesting applications, we need to know what it means by "two curves are orthogonal (perpendicular) to each other." (a) (1 mark) Let's begin with the simplest case: perpendicular lines. It is known that two lines on the plane are perpendicular to each other if either they are the pair of the r-axis and y-axis, or the products of their slopes are equal to -1. That is, if the two lines are defined by the following equations, y = mix + b1, y = max + b2, (m1, m2 # 0), they are perpendicular to each other if mim2 = -1 Now, find the equation of the line that is perpendicular to the line defined by y = 2x+ 1 and passes through the point (2, 1). (b) (2 marks) Using the concept of tangent lines, we can generalize the previous case to any curves on the plane that are meeting at a point P. Namely, we say such curves are orthogonal (perpendicular) at the point P if their tangent lines at the point P are perpendicular to each other. Justify that the following two circles are orthogonal to each other at a point which is not the origin. x2 +y? = 4x, x2 + y = 2y You may want to sketch the circles, but graphical justification will not be accepted for a grade. (c) (2 marks) Now, we make a further generalization: we say a curve C is orthogonal to a collection (family) of curves if C is orthogonal to every curve in this collection where they meet. Justify that the straight line y = x is orthogonal to the collection of all concentric circles defined by x2 +y? =12, where r is any positive real number. 2 You may want to sketch the circles and the line, but graphical justification will not be accepted for a grade