(See the Guided Projects for additional applications of conic sections.)Reflection property of parabolas Consider the parabola y = x2/(4p) with its focus at F(0, p) (see figure). We know that light is reflected from a surface in such a way that the angle of incidence equals the angle of reflection. The goal is to show that the angle between the ray ?and the tangent line L (? in the figure) equals the angle between the line PF and L (? in the figure). If these two angles are equal, then the reflection properly is proved because ? is reflected through F.a. Let P(x0, y0) be a point on the parabola. Show that the slope of the line tangent to the curve at P is tan ? = x0/(2p).b. Show that tan ?= (p ? y0)/x0.c. Show that ? = ?/2 ? ?; therefore, tan ? = cot ?.d. Note that ? = ? + ?. Use the tangent addition formula to show that tan ? = tan ? = 2p/x0.e. Conclude that because ? and ? are acute angles, ? = ?.
Solution 83EStep 1 of 5:a) In this problem we need to Show that the slope of the line tangent to the curve at P is tan(.Given parabola is y = with its focus at F (0 , p).Let p() be a point on the parabola.Let be an inclination of the tangent line , then the slope of the tangent line is tan(We know that ,slope is denoted by = m = tan(Consider , y = then differentiate both sides with respect to x we get , = () = , since cwhere c is constant. = .Therefore , the slope of the tangent line at the point p() is : = m = tan(= Step 2 of 5:b) In this problem we have to show that tan = .The given figure is ; Clearly , from the figure : tan() = -tan() = = = , since OF = , OB||OP = p and OP = .Therefore , tan() = .Step 3 of 5:c) In this we need to show that = /2 ; therefore, tan = cot .From the figure , it is clear that the line ‘l’ makes an angle with the horizontal line , and the tangent line ‘L’ makes angle with the horizontal line , and also the angle between the line ‘l’ and the tangent line ‘L’ is Therefore , - . - . = .Therefore , = .