Section 3: Geometry

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Algebra || Analysis

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1. Find the reflection of the point (2, 1) with respect to the line x=y in the xy-plane.
2. Find the area of the circle in the xy-plane which has its centre at the point (1,2) and which has the line x=y as a tangent.
3. Find the incentre of the triangle in the xy-plane whose sides are given by the lines x=0 , y=0 and $latex \frac {x}{3} + \frac {y}{4} = 1 $
4. Let A and B be fixed points in a plane such that, the length of the line segment AB is d. Let the point P describe an ellipse by moving on the plane such that the sum of the lengths of the line segments PA and PB is a constant l. Express the length of the semi-major axis, a and the length of the semi-minor axis, b, of the ellipse in terms of d and l.
5. Let A = $latex a_{ij} $ be a non-zero $latex 2 \times 2 $ symmetric matrix with real entries. Let  S = { $latex (x, y ) \in R^2 | a_{11} x^2 + 2 a_{12} xy + a_{22} y^2 = 0 $ }. Which of the following conditions imply that S is unbounded?
   1. det(A) > 0.
   2. det(A) = 0
   3. det(A) < 0.
6. Let $latex A \in M_2 (R) $  define an invertible linear transformation on $latex R^2 $. Let T be a triangle with one of its vertices at the origin and of area a. What is the area of the triangle which is the image of T under this transformation'?
7. Find the area of the ellipse whose equation in the xy plane is given by $latex 5 x^2 - 6 x y + 5 y^2 = 8 $.
8. Let a, b and c be positive real numbers. Find the equation of the sphere which passes through the origin and through the points where the plane $latex \frac {x}{a} + \frac {y}{b} + \frac {z}{c} $ meets the coordinate axes.
9. Consider the sphere $latex x^2 + y^2 + z^2 = r^2 $.  Let $latex (a, b, c ) \ne (0, 0, 0) $ be a point in the interior of this sphere. Write down the equation of the plane whose intersection with the sphere is a circle whose center is the point (a, b, c) .
10. Find the area of the polygon whose vertices are represented by the eighth roots of unity.

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Algebra || Analysis

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