Let the base AB of a triangle ABC be fixed and the vertex C lies on a fixed circle of radius r. Lines through A and B are drawn to intersect CB and CA, respectively, at E and F such that CE: EB=1:2 and CF: FA= 1:2.If the point of intersection P of these lines lies on the median through AB for all positions of AB then the locus of P is.
a circle of radius r2
a circle of radius 2r
a parabola of latus rectum 4r
a rectangular hyperbola
A
a circle of radius r2
B
a circle of radius 2r
C
a rectangular hyperbola
D
a parabola of latus rectum 4r
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Solution
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Consider the circle in which the triangle is inscribed with bare AB. The vertex C of △ABC is fixed at the centre of the circle of radius r.
A line is drawn through A and it intersects CB at E and dividing CB in the ratio CE: EB = 1:2. Similarly, a line is drawn through B and it intersects AC at F and divides AC in the ratio 1:2. Let P be the centroid i.e. P (h, k)
Let the base AB of a triangle ABC be fixed and the vertex C lies on a fixed circle of radius r. Lines through A and B are drawn to intersect CB and CA, respectively, at E and F such that CE: EB=1:2 and CF: FA= 1:2.If the point of intersection P of these lines lies on the median through AB for all positions of AB then the locus of P is.
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Q2
Let these base AB of a triangle ABC be fixed and the vertex C lie on a fixed circle of radius r. Lines through A and B are drawn to intersect CB and CA, respectively, at E and F such that CE:EB=1:1 and CF:FA=1:2. If the point of intersection P of these lines lies on the median through AB for all positions of AB then the focus of P is
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Q3
The base AB of a triangle ABC is fixed and vertex C moves such that sinA=ksinB. If AB=5 then
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Q4
AB is a fixed line. State the locus of the point P so that AB2= AP2+ BP2
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Q5
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