Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE Main or Advanced/AIEEE, and anyone else who needs this Tutorial as a reference! After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well. The Ellipse: Equation of an Ellipse, Eccentricity, Focus, Major and Minor Axis, Chords and Latus Rectum:
Important Formulas and EquationsParametric Form, Tangents, Normals, Chords and Diameter of an EllipseNote : The results are for the standard form of ellipse! Parametric Form Parametric form of ellipse is (acosθ,bsinθ) Tangentxx1/a2+yy1/b2=1 , at point (x1,y1) y=mx±a2m2+b2, the point of contact being (±a2ma2m2+b2,±b2a2m2+b2) or (xcosθ)a+(ysinθ) b=1, the point of contact being (acosθ,bsinθ) From an external point, two tangents can be drawn to the ellipse. Normalsa^{2}x-x1x1=b^{2}y-y1y1, point of contact being x1,y1, upon simplification a2xy1-b2yx1-a2-b2x1y1=0 axcosθ+bysinθ=a2-b2, the point of contact being (acosθ,bsinθ) From a fixed pint, four normals can be drawn to the ellipse. ChordEquation of chord joining (acosα,bsinα) and (acosβ,bsinβ) is xacosα+β2+y2sinα+β2=cosα-β2 *The locus of the point of intersection of two perpendicular tangents to an ellipse is a circle known as the director circle. *Auxiliary circle of an ellipse is a circle which is described on the major axis of an ellipse as its diameter. *Diameter. The locus of mid points of a system of parallel chords is called the diameter. *Conjugate Diameters. Two diameters are conjugate diameters which bisects chords parallel to each other. Condition turns out to be mm1=-b2/a2 The only pair of conjugate diameters that are perpendicular to each other and does not satisfy the above condition is the major and minor axis of ellipse. If two conjugate diameters are equal, then they are called equi-conjugate diameters. . Here are some of the problems solved in this tutorial :Q: Find the equation of ellipse with focus at (1,1), eccentrity ½ and directrix x-y+3=0. Also, find the equation of its major axis. Q: Find lengths of major and minor axes, coordinates of foci and vertices, and the eccentrity of x2+4y2-2x=0. Q: A man running a race course notes that the sum of distances from the two flag posts is always 10 metres and the distance between the flag posts is 8 metres. Find the equation of the path traced by him. Q: Find the locus of the point of intersection of the tangents which meet at right angles. Q: Prove that the product of the lengths of perpendiculars drawn from foci to any tangent is b2. (a>b) Q: If the normals at the end of a latus rectum passes through the extremity of a minor axis, prove that e4+e2=1. Q: Find the chord of contact wrt point (x1,y1). Q: Find the equation of a pair of tangents drawn from P(x1,y1) Q: Prove that the common tangent of the circle x2+y2=4 and x27+y23=1 is inclined to major axis at an angle of 30 degrees. Q: Find the equation of a chord of the ellipse whose mid point is (x1,y1) Complete Tutorial (There is an MCQ Quiz below it)MCQ Quiz/Worksheet on Ellipses. |