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 .raxis and ;;axis, or the products of their slopes are equal to 1. That is: if the two lines are dened by the following equations, 3; = mm + b1, 1; = mg: + 52.- [nt.._ 1112 ;E [J]. they are perpendicular to each other if mung = 1 Now: nd the equation of the line that is perpendicular to the line dened by y = 2.15 l l and passes through the point [2, 1}. [h] (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. Justifyr that the following two circles are orthogonal to each other at a point which is not the origin. 12+y2=4x, 12+y2=2y You may want to sketch the circles. but graphical justication will not be accepted for a yade. {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 :1; = 1' is orthogonal to the collection of all concentric circles dened by r2 + y2 = r24 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