Mathematics Ellipse

8/12/2019 Mathematics Ellipse

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PROBLEMS 05 - ELLIPSE Page 11 ) The endopolnts A and B of AB are on the X and Y-axls respectively. If AB = a + b,

a > 0, b > 0, a b AB from A in the ratio b : a, then show thatP lies on the ellipse

2 I the perpendiculars drawn tofrom foci S and S are

3 I

41

5 1 If the chord j o i n i n ~ ~ a ) and Q p1 of the ellipse0 a II e 1through the foe , then prove that t an t an - 2-e 1 1 passesthe points P a, ) and Q { p I of the ellipse x2 y2 = 1a2 b2

a2en s a right angle at the centre, then show that tan a, tan p + b =0 and i f ip b2orms a right angle at the vertex a, 0 ), then show that tan a t an + 2 =0.2 2 a

7 ) If the difference of eccentric angles of the points P and Q on the ellipsex2 y2- + - = 1 Is and PQ cuts Intercepts of length c and d on the axes, thena2 b2 2

a2prove that - +c:


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PRO LEMS 05 - ELLIPSE Page 2

8 ) If two radii CP and CQ of the ellipse x2 y2- + - = 1 are perpendicular, then provea2 b2

9 )

that 1 +cp2 1ca 2 = 1 1+ where C is the centreb2.Find the equations of the tangents drawn to the ellipse, 9x2 Ql. 144 from the

point 2, 3 ). ~[ Ans: x + y 5 =0 y 3 =0 ]

14 ) P and Q are coherent points on the ellipse x2 y2- + - = 1 and its auxiliary circle.a2 b2Show that the tangents to the ellipse at P and the circle at Q intersect on the X-axls.a > b


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PROBLEMS 05 - ELLIPSE Page 3( 15 ) If the lengths of the perpendicular line-segments from the centre to two mutually

x2 y2orthogonal tangents of the ellipse - + - = 1 are P and P2, then prove thaa2 b22 z z 2Pt p2= a b.

16 ) A tangent to the ellipse Intersects the axes

( 17)

( 18)

and touches the ellipse at mid-point of CO In the first qua

ZJIf the perpendicular distance

. x2 y2 .ellipse - + - =1 sa2 b2of the fo ur t iue tangent at a point p on the

_ 2ap2p, then pr JI. th - 2 2 .b P

x2 y2- + - =1 a > b ) passinga2 b2through f o u s ~ another end-point, then show that the slope of CP Is(1 - e2)2 ~ ~ centre of the ellipse.

2e CJ19 ) Th fo he perpendicular from a point P on ellipse to the major axis Is M. ISP.

' _, 1 intersects the auxiliarycircle In Q and Q , then prove that SQ x SQ =

2 ) Prove that if the tangent at a point P to the ellipse intersects a directrix at F, thenPF forms a right angle at the corresponding focus.


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PROBLEMS 05 - ELLIPSE P a g ~22 ) The tangent at point P of an ellipse l n t ~ r s e c t s the major axis In T. The line passing

through T and perpendicular to major axis AA , Intersectsand Q respectively. Show that T Is the mid-point of QQ.

AP and A P In Q

23 ) The tangent at point P of an ellipse x2 Trespectively. If R Is the foot ofprove that TT PR =a2 - b2. perpendicular from the centJto e tangent. then

( 24 ) Find the condition that the line x + my + n a tangent to the el l ipux2 y2- + - = 1 and find thea2 b2

a 2f ,+ b2m2 =n2, n nx2 12- + - = 1 with centre C meets thea2 b2( 25 ) I f the tangent at any

a2 b2major axis In T a lno s In T . then prove that + = 1 ( a > b ).CT 2 CT 2( 26 ) Find the conde. the line x cos a + y sin a = p to be a tangent to the ellipse

x2 y2 1.+a2 cot2 a + b2 )s and Q are corresponding points on an e l l l p s ~ and Its auxiliary~ - the tangent at P to the ellipse meets the maJor axis In T, thentangent to the auxiliary circle. c l r c l ~ respectively. I< -+show that QT is a

( 28 Find the perpendicular distance between the t a n g ~ n t s to the ellipse .2 + = 130 24which are parallel to the line 4x - 2y + 23 = 0.

Ans: 24 J5 )


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PRO LEMS 5 - ELLIPSE P a g ~ > 5

( 29 ) Prove that the equation of the chord of the ellipse

point at ( h, k ) Is :: + = :: + ::( 30 ) Prove that the equation of the chord joining the points P ( a Q a P on

x2 y2 x ythe ellipse + b2 =1 Is a os a + bs lna =cosP, . ( 31 ) Prove that the area of the triangle formed by t ( 8 ), Q ( a ) and R ( p ) on

x2 y2 Ithe ellipse 2 + 2 = 1 is 2aba b( 32 ) Prove that the equations of t

a Iln-2- .

and the

circle x2 + y2 = r2 ~' I . . . . . . of t i l t ~ - ...... . . . . . . . . of ..............of the ellipse. e hat the locus of their centres Is he curve ~ , . ., , .., .( ' mnooN of ' ' ' ~ - ' ' ,., ho -

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Mathematics Ellipse