NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y

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1 LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to e norml to this ellipse. The slope of the tngent t P cn e found evluting the derivtive d d Thus, t P. m T d d P P Therefore, the slope of the norml is m N The eqution of the norml cn now e written using point-slope form: ( ) If P hs een specified in prmetric form, the eqution for the norml cn e otined the sustitution cos, sin in the eqution otined ove : sec cosec This form of the norml is the most widel used. It represents the norml t point whose eccentric ngle is. Let m c e norml to the ellipse (s t the point P(, )). The eqution of the norml t P(, ) cn e written s...() This hs slope m...() Mthemtics / Ellipse

2 LOCUS 5 Since P(, ) lies on the ellipse, we hve Using () in (3), we otin m 4...(3) m...(4) m From (),...(5) m Using (4) nd (5) in (), we finll otin the eqution of the norml with slope m s m m ( ) m We cn s tht n line of the form (6) will e norml to the ellipse the points of contct given (4) nd (5)....(6), with Emple 3 When is the stright line p q r 0 norml to the ellipse? An norml to the ellipse cn e written using the prmetric form s If is lso norml to the ellipse, we hve sec cosec. p q r sec cosec r P q r cos Eliminting gives us the required condition : r r, sin p q cos sin ( ) ( ) ( ) r p q Mthemtics / Ellipse

3 LOCUS 5 Emple 3 Wht is the frthest distnce t which norml to the ellipse cn lie from the centre of the ellipse? An norml to the ellipse is of the form sec cosec The distnce of this norml from the centre (0, 0) is d sec cosec We need to find the mimum vlue of d, or equivlentl, the minimum vlue of We hve f ( ) sec cosec f ( ) sec tn cosec cot f ( ) 0 when tn tn Verif tht t this vlue of, f ( ) is positive so tht this indeed gives us the minimum vlue of f ( ). Now, fmin ( ) sec min cosec min ( tn ) ( cot ) min ( ) min d m f ( ) min Mthemtics / Ellipse

4 LOCUS 53 Emple 33 Find the point on the ellipse 6 3 whose distnce from the line 7 is minimum. An point on the given ellipse cn e ssumed to e P ( 6 cos, 3 sin ). From the following figure, oserve tht for the distnce of P from the given line to e minimum, the norml t P must e perpendiculr to the given line. P Q If P is the point of the minimum distnce from the given line, the norml t P must e perpendiculr to the given line. The eqution of the norml t P, using prmetric form, is whose slope is ( 6 sec ) ( 3cosec ) 3 m N tn If the norml is perpendiculr to 7, we hve m N tn sin 3 nd cos 3 Thus, the point P is ( cos, sin ) (,). Mthemtics / Ellipse

5 LOCUS 54 Emple 34 The norml t n point P on the ellipse meets the mjor nd minor es t A nd B respectivel. ON is the perpendiculr upon this norml from the centre O of the ellipse. Show tht PA PN nd PB PN M P O N A B Assume the point P to e ( cos, sin ). The norml t P hs the eqution The coordintes of A re therefore Similrl, B is ( sec ) ( cosec )...() A B sec,0 ( e cos,0) e 0, 0, sin cosec PA nd PB cn now e evluted using the distnce formul : PA ( cos e cos ) ( sin ) 4 cos sin sin cos...() e ( cos ) sin sin PB Mthemtics / Ellipse sin cos...(3)

6 LOCUS 55 PN cn e evluted either using the perpendiculr distnce of O from the norml t P( PN OP ON ) or simpl s the perpendiculr distnce of O from the tngent t P. The tngent t P hs the eqution Thus, cos sin 0 PN From (), (3) nd (4), we hve cos sin...(4) PA PN nd PB PN Emple 35 Prove tht from n given point P( h, k ), four normls (rel or imginr) cn e drwn to the ellipse nd the sum of the eccentric ngles of the feet of these four normls is n odd integrl multiple of. An norml to the ellipse is of the form If this psses through P( h, k ), we hve ( sec ) ( cosec ) hsec k cosec...() We need to show tht this eqution will in generl ield four vlues of. For this purpose, we use the sustitution cos t, sin t t t where t tn. Thus, () trnsforms to, t t h k e t t ( ( e )) 4 3 kt ( h e ) t ( h e ) t k 0...() Mthemtics / Ellipse

7 LOCUS 56 This is iqudrtic eqution in t, ielding four roots, s t, t, t 3, t 4. This shows tht in generl four normls cn e drwn. From (), we hve s t t t t 3 4 ( h e ) k s tt tt 3 tt 4 tt3 tt4 t3t4 0 s t t t t t t t t t t t t ( h e ) k Thus, s k k 4 tt t3t4 3 4 s s3 tn s s 4 n (n ) This proves tht the sum of the eccentric ngles is n odd multiple of. Mthemtics / Ellipse

8 LOCUS 57 TRY YOURSELF - IV The norml t n point P on n ellipse cuts its mjor is in Q. Show tht the locus of the mid-point of PQ is n ellipse. If the eccentric ngles of points A nd B on the ellipse re nd nd is the ngle etween the normls t A nd B, prove tht the eccentricit e of the ellipse is given e sin tn. e The tngent drwn t the point ( t, t ) on the prol 4 is the sme s the norml drwn t point ( 5 cos, sin ) on the ellipse Wht re the vlues of t nd? If the norml t n end of ltus rectum of n ellipse psses through re etremit of the minor is, show tht its eccentricit e stisfies 4 e e. A r is incident on the ellipse t point with -coordinte 4. The source of this r is t ( 3, 0). Find the eqution of the reflected r. Prove tht the norml t n point on n ellipse isects the ngles etween the focl rdii of tht point. Mthemtics / Ellipse

Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE 2 1 = 0 Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).