11%20-%20Two%20Dimensional%20Geometry

11%20-%20Two%20Dimensional%20Geometry - 11 - TWO...

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11 - TWO DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) ( 1 ) Area of the greatest rectangle that can be inscribed in an ellipse 2 2 2 2 b y a x + = 1 is ( a ) 2ba ( b ) ab ( c ) ab ( d ) b a [ AIEEE 2005 ] ( 2 ) Let P be the point ( 1, 0 ) and Q the point on the locus y 2 = 8x. The locus of midpoint of PQ is ( a ) y 2 - 4x + 2 = 0 ( b ) y 2 + 4x + 2 = 0 ( c ) x 2 + 4y + 2 = 0 ( d ) x 2 - 4y + 2 = 0 [ AIEEE 2005 ] ( 3 ) The line parallel to the X-axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx - 2ay - 3a = 0, where ( a, b ) ( 0, 0 ) is ( a ) below the X-axis at a distance 2 3 from it ( b ) below the X-axis at a distance 3 2 from it ( c ) above the X-axis at a distance 2 3 from it ( d ) above the X-axis at a distance 3 2 from it [ AIEEE 2005 ] ( 4 ) The locus of a point P ( α , β ) moving under the condition that the line y = α x + β is a tangent to the hyperbola 2 2 2 2 b y a x - = 1 is ( a ) an ellipse ( b ) a circle ( c ) a parabola ( d ) a hyperbola [ AIEEE 2005 ] ( 5 ) If non-zero numbers a, b, c are in H.P., then the straight line a x + b y + c 1 = 0 always passes through a fixed point. That point is ( a ) ( - 1, 2 ) ( b ) ( - 1, - 2 ) ( c ) ( 1, - 2 ) ( d ) ( 1, - 2 1 ) [ AIEEE 2005 ] ( 6 ) If a vertex of a triangle is ( 1, 1 ) and the midpoint of two sides through this vertex are ( - 1, 2 ) and ( 3, - 2 ), then the centroid of the triangle is ( a ) ( - 1, 3 7 ) ( b ) ( - 3 1 , 3 7 ) ( c ) ( 1, 3 7 ) ( d ) ( 3 1 , 3 7 ) [ AIEEE 2005 ]
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11 - TWO DIMENSIONAL GEOMETRY Page 2 ( Answers at the end of all questions ) ( 7 ) If the circles x 2 + y 2 + 2ax + cy + a = 0 and x 2 + y 2 - 3ax + dy - 1 = 0 intersect in two distinct points P and Q, then the line 5x + by - a = 0 passes through P and Q for ( a ) exactly one value of a ( b ) no value of a ( c ) infinitely many values of a ( d ) exactly two values of a [ AIEEE 2005 ] ( 8 ) A circle touches the X-axis and also touches the circle with centre at ( 0, 3 ) and radius 2. The locus of the centre of the circle is ( a ) an ellipse ( b ) a circle ( c ) a hyperbola ( d ) a parabola [ AIEEE 2005 ] ( 9 ) If a circle passes through the point ( a, b ) and cuts the circle x 2 + y 2 = p 2 orthogonally, then the equation of the locus of its centre is ( a ) x 2 + y 2 - 3ax - 4by + ( a 2 + b 2 - p 2 ) = 0 ( b ) 2ax + 2by - ( a 2 - b 2 + p 2 ) = 0 ( c ) x 2 + y 2 - 2ax - 3by ( a 2 - b 2 - p 2 ) = 0 ( d ) 2ax + 2by - ( a 2 + b 2 + p 2 ) = 0 [ AIEEE 2005 ] ( 10 ) An ellipse has OB as semi minor axis, F and F’ its foci and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is ( a ) 2 1 ( b ) 2 1 ( c ) 4 1 ( d ) 3 1 [ AIEEE 2005 ] ( 11 ) If the pair of lines ax 2 + 2 ( a + b ) xy + by 2 = 0 lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector, then ( a ) 3a 2 - 10ab + 3b 2 = 0 ( b ) 3a 2 - 2ab + 3b 2 = 0 ( c ) 3a 2 + 10ab + 3b 2 = 0 ( d ) 3a 2 + 2ab + 3b
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This note was uploaded on 11/28/2011 for the course MATH 300 taught by Professor Jones during the Spring '06 term at ITT Tech Flint.

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11%20-%20Two%20Dimensional%20Geometry - 11 - TWO...

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