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Find the reflection of the point (2, 1) with respect to the line x=y in the xy-plane.

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.

Find the incentre of the triangle in the xy-plane whose sides are given by the lines x=0 , y=0 and

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.

Let A = be a non-zero symmetric matrix with real entries. Let S = { }. Which of the following conditions imply that S is unbounded?

det(A) > 0.

det(A) = 0

det(A) < 0.

Let define an invertible linear transformation on . 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’?

Find the area of the ellipse whose equation in the xy plane is given by .

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 meets the coordinate axes.

Consider the sphere . Let 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) .

Find the area of the polygon whose vertices are represented by the eighth roots of unity.

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