Chapter 12. Parabolas Definitions

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1 Chapter 12 Parabolas 12.1 Definitions Given a point (focus) and a line (directrix), the locus of a pointp which is equidistant from and is a parabolap. et the distance between and be 2a. Set up a Cartesian coordinate system such that = (a,0) and has equation x = a. The origin is clearly on the locus. Its distances from and are botha. Q P(x,y) directrix a = (a,0) et P(x,y) be a point equidistant from = (a,0) and. Then (x a) 2 + y 2 = (x+a) 2. Rearranging terms, we have y 2 = 4ax.

2 302 Parabolas The parabola P can also be described parametrically: { x = at 2, y = 2at. We shall refer to the point P(t) := (at 2, 2at) as the point on P with parameter t. It is the intersection of the line y = 2at and the perpendicular bisector of Q, Q = ( a,2at) on the directrix Chords and tangents The line joining the points P(t 1 ) and P(t 2 ) has equation 2x (t 1 +t 2 )y +2at 1 t 2 = 0. etting t 1, t 2 t, we obtain the equation of the tangent at P(t): x ty +at 2 = 0. The tangent at P(t) has slope 1 t and intersects the axis of the parabola at ( at2, 0). It can be constructed as the perpendicular bisector of Q, where Q is the pedal of P on the directrix. Exercise 1. (ocal chord) The line joining two distinct points P(t 1 ) and P(t 2 ) on P passes through the focus if and only if t 1 t 2 +1 = (Intersection of two tangents) Show that the tangents at the points t 1 and t 2 on the parabola y 2 = 4ax intersect at the point (at 1 t 2, a(t 1 +t 2 )). 3. Justify the construction suggested by the diagram below for the construction of the tangent at a point P on a parabola. P

3 Q 2 T Chords and tangents 303 T 1 Q 1 P 4. (Tangents from a point to a parabola) Given a point P, construct the circle with diameter P and let it intersect the tangent at the vertex at Q 1 and Q 2. Then the lines PQ 1 andpq 2 are the tangents from P to the parabola. 5. et P 1 P 2 be a focal chord of a parabola P with midpoint M. The tangents at P 1 and P 2 intersect atq. Show that (i) Q lies on the directrix, (ii) these tangents are perpendicular to each other, and (iii) QM is parallel to the axis of the parabola, and (iv) QM and intersects P at its own midpoint. 6. Justify the following construction of the chord of a parabola which has a given point M as its midpoint: et Q be the pedal of M on the directrix. Construct the circle M(Q) to intersect the perpendicular from M to Q. These two intersections are on the parabola, and their midpoint is M. P 1 Q M P 2 7. A circle on any focal chord of a parabola as diameter cuts the curve again at two

4 304 Parabolas points P and Q. Show that as the focal chord varies, the line PQ passes through a fixed point. 8. (Chords orthogonal at the vertex) et PQ be a chord of a parabola with vertex such that angle PQ is a right angle. ind the locus of the midpoint of PQ. 9. ind the locus of the point whose two tangents to the parabola y 2 = 4ax make a given angle α Normals The normal atp(t) of the parabolay 2 = 4ax is the perpendicular to the tangent at the same point. It has slope t; its equation is tx+y (2at+at 3 ) = 0. It intersects the parabola again atp(t ), where t = t 2 t. P Q t

5 12.4 Triangle bounded by three tangents 305 Exercise 1. (Reflection property) Prove the following reflection property of the parabola. et P be a point on a parabola with focus. The reflection of the focal chord P in the normal atp is parallel to the axis of the parabola. P 2. (rthogonal normals) ind the condition for the normals at the points t 1 and t 2 on the parabola y 2 = 4ax to be orthogonal. Hence, find the locus of the intersection of the orthogonal normals. 3. (Normals intersecting on parabola) ind the condition ont 1 andt 2 so that the normals to the parabola y 2 = 4ax at the points t 1 and t 2 intersect on the parabola. 4. Show that if the normals at A and B on the parabola y 2 = 4ax intersect at a point C on the parabola, then (a) the tangents ataandb intersect at a point on the line x = 2a, (b) the centroid of the triangle ABC lies on the axis of the parabola. 5. (Three concurrent normals) Show that the normals at the points P(t 1 ), P(t 2 ), P(t 3 ) on the parabola y 2 = 4ax are concurrent if and only if t 1 +t 2 +t 3 = The normals at three points A, B, C on a parabola are concurrent. Show that the centroid of triangle ABC lies on the axis of the parabola. 7. PQ is a focal chord of a parabola with focus. Construct the circles through tangent to the parabola at P and Q. What is the locus of the second intersection of the circles (apart from )? 12.4 Triangle bounded by three tangents Proposition The points P(t 1 ), P(t 2 ), P(t 3 ), P(t 4 ) on the parabola y 2 = 4ax are concyclic if and only if t 1 +t 2 +t 3 +t 4 = 0. Theorem The circumcircle of the triangle bounded by three tangents to a parabola passes through the focus of the parabola.

6 306 Parabolas P 1 A 3 3 P 2 A 2 A P 3 Proof. or i = 1,2,3, the tangent at t i is the line i = x t i y +at 2 i = 0. We find λ 1, λ 2, λ 3 such that λ λ λ = 0 represents a circle. This requires the coefficients of x 2 and y 2 to be equal, and that of xy equal to 0: (1 t 2 t 3 )λ 1 + (1 t 3 t 1 )λ 2 + (1 t 1 t 2 )λ 3 = 0, (t 2 +t 3 )λ 1 + (t 3 +t 1 )λ 2 + (t 1 +t 2 )λ 3 = 0. Solving these equations, we have λ 1 : λ 2 : λ 3 = 1 t 3t 1 1 t 1 t 2 t 3 +t 1 t 1 +t 2 : 1 t 1 t 2 1 t 2 t 3 t 1 +t 2 t 2 +t 3 : 1 t 2 t 3 1 t 3 t 1 t 2 +t 3 t 3 +t 1 = (t 2 t 3 )(1+t 2 1) : (t 3 t 1 )(1+t 2 2) : (t 1 t 2 )(1+t 2 3). With these values of λ 1,λ 2,λ 3, we compute the common coefficient of x 2 andy 2 as (t2 t 3 )(1+t 2 1) = (t 2 t 3 )(t 3 t 1 )(t 1 t 2 ). Also, coefficient of x = (t 2 t 3 )(1+t 2 1) a(t 2 2 +t 2 3) = a(t 2 t 3 )(t 3 t 1 )(t 1 t 2 )(1+t 2 t 3 +t 3 t 1 +t 1 t 2 ), coefficient of y = (t 2 t 3 )(1+t 2 1) a(t 2 2 +t 2 3) = a(t 2 t 3 )(t 3 t 1 )(t 1 t 2 )(t 1 +t 2 +t 3 t 1 t 2 t 3 ), the constant term = (t 2 t 3 )(1+t 2 1) a 2 t 2 2t 2 3 = a 2 (t 2 t 3 )(t 3 t 1 )(t 1 t 2 )(t 2 t 3 +t 3 t 1 +t 1 t 2 ).

7 12.5 Evolute of a parabola 307 Cancelling a common factor (t 2 t 3 )(t 3 t 1 )(t 1 t 2 ), we obtain the equation of the circle as x 2 +y 2 a(1+σ 2 )x a(σ 1 σ 3 )y +a 2 σ 2 = 0, where σ 1,σ 2, σ 3 are the elementary symmetric functions of t 1, t 2,t 3. This circle clearly passes through the focus = (a,0). Exercise 1. Prove that the orthocenter of the triangle formed by three tangents to a parabola lies on the directrix. 2. NormalsPX,PY,PZ are drawn to a parabola, focus. Prove that the reflection of P in lies on the circle circumscribing the triangle formed by the tangents atx,y, Z Evolute of a parabola The normals to the parabola P : y 2 = 4ax at the points t 1 and t 2 intersect at the point (2a+a(t 2 1+t 1 t 2 +t 2 2, at 1 t 2 (t 1 +t 2 )). By puttingt 1 = t 2 = t, we obtain the intersection of the normal atp and its immediate neighbor, namely, Q(t) = (2a+3at 2, 2at 3 ). The circle throughp(t), with centerq(t), is the limiting position of a circle throughp and two immediate neighbors. This is called the circle of curvature of the parabola atp. P T N M K Suppose the tangent and normal at P intersect the axis at T and N respectively. Construct (1) the parallel throught to the normal, to intersect the ordinate throughp (perpendicular to the axis) atm, (2) the parallel through M to the axis, to intersect the normal atk. The point K is the center of curvature of the parabola atp.

8 308 Parabolas The equation of the circle of curvature attis x 2 +y 2 2(2a+3at 2 )x+4at 3 y 3a 2 t 4 = 0. The radius of curvature is 2a(1+t 2 ) 3 2. The locus of the center of curvature is called the evolute of the parabola. It is the semicubical parabola 27ay 2 = (x 2a) 3. P(t) 2a Q(t) Exercise 1. ind the minimum normal chord of the parabola y 2 = 4ax. 2. Show that the normal to the parabola y 2 = 4x at the point t is tangent to its evolute and intersects it again at Q ( t 2) (apart from Q(t)). P(t) P( t 2a 2 ) Q( t 2 ) Q(t) 3. (Pedal curve) etabe a fixed point. or a variable pointp on the parabolay 2 = 4ax, letqbe the pedal of A on the tangent atp. (a) ind a parametrization of the locus of Q. (b) Describe this locus with reference to the evolute of the parabola.

9 12.6 Miscellaneous exercises Miscellaneous exercises 1. ind the equation of the perpendicular bisector ofp and show that it envelopes the parabola. 2. et a > 0 be a constant. Show that the line x + y t locate the focus and the directrix. a t = 1 envelopes a parabola, and 3. (Parabolas with a common vertex and perpendicular axes) The two parabolas P : y 2 = 4ax andp : x 2 = 4by have a common vertex and perpendicular axes. eta be their common point other than. The tangent atatopintersectsp atb and the tangent at A to P intersects P at C. Show that the line BC is a common tangent of the parabolas. 4. (Conformal focal parabolas) Two parabolas have a common focus ; their directrices intersects at a point A. Show that the perpendicular bisector of A is the common tangent of the two parabolas. 5. Tangents drawn to two confocal parabolas from a point on the common tangent intersect at the same angle as the axes of the parabolas. 6. (a) ind the condition for the two parabolas y 2 = 4ax and y 2 = 4b(x c) to have a common tangent. (b) Construct the common tangent of the two parabolas. 7. etabe a point on a circle(), andtis the tangent ata. TrianglePQ is such that P is on(),qis ont, and PQ = 90. ind the envelope of the perpendicular to AP through Q as triangle P Q varies. 8. rom a point P normals are drawn to a parabola. ind the locus of P so that the intersections with the parabola form a right triangle.

10 310 Parabolas Appendix: center of curvature Consider the normal of a parametric curve C : x = f(t), y = g(t) at the point t, with equation given by f (t)(x f(t))+g (t)(y g(t)) = 0. At a neighboring pointt, the normal is given by the same equation withtreplaced byt. If we writet = t+ε for a smallε, and replacef(t ) byf(t)+f (t)ε,f (t) byf (t)+f (t)ε, and similarly for g(t) andg (t), then the normal at this neighboring point is the line f (t)(x f(t))+g (t)(y g(t)) + ε(f (t)(x f(t))+g (t)(y g (t)) = 0. Therefore, solving the system of equations f (t)(x f(t))+g (t)(y g(t)) = 0, f (t)(x f (t))+g (t)(y g (t)) = 0, we obtain the intersection of two neighoring normals, namely, { x = f(t) g f (t) (t) 2 +g (t) 2, f (t)g (t) f (t)g (t) y = g(t)+f f (t) (t) 2 +g (t) 2. (12.1) f (t)g (t) f (t)g (t) This is called the center of curvature at P(t). With this point as center, the circle through P is tangent to the curve C. The radius of this circle is (f (t) 2 +g (t) 2 ) 3 2 ρ = f (t)g (t) f (t)g (t). The curvature atp(t) is κ := 1 ρ.

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola) QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents