y u; 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 xaxis and yaxis, or the products of their slopes are equal to L That is, if the two lines are dened by the following equations, SI' = W113+ b1: SI' = m2x+ 52: (\"7-1me 9E 0): they are perpendicular to each other if mlmg = 1 Now, nd the equation of the line that is perpendicular to the line dened by y = 2x + l and passes through the point (2, l)' (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 Iines 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. 2 2 _ 2 2 _ m +y 42, :5 +3.; 2y You may want to sketch the circles, but graphical justication will not be accepted for a grade; (c) (2 marks) Now, we make afurt-her 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 = a: is orthogonal to the collection of all concentric circles dened by 2:2 + y? = r2, where r is any positive real number. 2 You may want to sketch the circles and the line, but graphical justication will not be accepted for a grade