Objective Type Questions
1. The vertices of a triangle are (2, 1), (5, 2) and (3, 4) then circumcenter of the triangle
2. The co-ordinates of base BC of an isosceles triangle ABC are given by B(1, 3) and
C (-2, 7). Which of the following points can be the possible coordinates of the vertex A?
3. 'P' and 'Q' are points on the line joining A(-2, 5) and B(3, 1) such that AP = PQ = QB, then the mid point of PQ is
4. If distance between (a, 2) and (3, 4) is 8 then a =
5. The coordinates of incentre and centroid of the triangle whose vertices are (-36, 7), (20, 7), (0, -8)
A: (-1, 0)
6. If "G" be the centroid of a triangle ABC and "O" be any other point, then
7. In a ∆ ABC, if 'O' is mid point of 'BC', then
8. The locus of a point P(x, y) moves such that the sum of its distances from two fixed points (ae, 0) and (-ae, 0) is always 2a, is
9. The ends of a rod of length "l" move on two mutually perpendicular lines. The locus of the point on the rod which divides it in ratio 1 : 2 is
10. If and (p, q) are the co-ordinates of circumcenter, centroid and orthocentre of a trinagle, then
A: 2(α + β)
11. If P, Q, R divides the sides of the triangle ABC in the same ratio, then which of the following coincide for the triangles ABC, PQR?
12. If P(1, 2), Q(4, 6), R(5, 7) and S(a, b) are the vertices of a parallelogram PQRS, then
A: a = 2, b = 3
13. If the vertices P, Q, R of a trinagle PQR are rational points. Which of the following points of the triangle PQR is (are) always rational points?
A: All of these
14. If P be any point in the plane of square ABCD, then PA2 + PB2 =
A: PC2 + PD2
15. If the point (x, y) is equidistant from the points (a + b, b - a) and (a - b, a + b), then the value of
16. The equation of the locus of a point which moves so that the sum of their distances from (3, 0) and (-3, 0) is less than 9 is
A: 20x2 + 36y2 < 405
17. The locus of a point whose co-ordinates are given by x = t + t2 and y = 2t + 1 (where "t" is variable) is
A: y2 = 4x + 1
18. The locus of the point of intersection of the lines x cos α + y sin α = a and x sin α - y cos α = b (where "α" is variable) is
A: x2 + y2 = a2 + b2
19. A and B are fixed points, the vertex 'C' of ∆ABC moves such that cot A + cot B = constant. The locus of C is a straight line which is
A: parallel to AB
20. If the sum of the distances of a point from two perpendicular lines in the plane is "1", then its locus is
A: a straight line
21. If the distance of any point P(x, y) from the origin is defined as then the locus of "P" is
A: a straight line
22. If A(cos α, sin α), B(sin α, −cos α), C(1, 2) are the vertices of a ∆ ABC (where α is variable), then locus of centroid is
A: 3(x2 + y2) − 2x − 4y + 1 = 0
23. Let A(2, −3) and B(−2, 1) be the vertices of a triangle ABC. If the centroid of the triangle moves on the line 2x + 3y = 1, then the locus of the vertex 'C' is
A: 2x + 3y = 9
24. If the origin is shifted to the point (1, -2) without rotation, then the equation x2 + y2 − 4x − 4y becomes
A: 2x2 + y2 = 6
25. To what point the origin is to be shifted so that the equation y2 + 4y + 8x − 2 = 0 will not contain term in "y" and the constant term?
26. Through what angle should the axes be rotated so that the equation 9x2 − 2xy + 7y2 = 10 may be changed to 3x2 + 5y2 = 5?
27. If the axes be turned through an angle tan−1 (2), what does the equation 4xy − 3x2 = a2 become
A: x2 − 4y2 = a2
28. If (x, y) and (X, Y) be the coordinates of the same point referred to two sets of rectangular axes with the same origin. If ax + by becomes pX + qY where a, b are independent of x, y then a2 + b2 =
A: p2 + q2
29. What does the equation (x − a)2 + (y − b)2 = r2 become when the axes are transferred to parallel axes through the point (a − c, b)?
A: x2 + y2 − 2cx = r2 − c2
30. If by rotating the coordinate axes without translating the origin, the expression a1x2 + 2h1xy + b1y2 becomes a2x2 + 2h2xy + b2y2, then which of the following is wrong?
A: (a1b2 + a2b1)2 = h12 + h22
31. The point (4, 1) undergoes the following transformation successively.
(i) Reflection about the line y = x
(ii) Transformation through a distance 2 units along the positive direction of X−axis
(iii) Rotation through an angle Π/4 about the origin in anti clockwise direction
Final position of the point is given by coordinates is
32. A(2, 1), B(3, −7) are two points. C is any point on the line 3x − 2y = 1, then locus of point "D" such that ABCD is a parallelogram
A: 3x − 2y + 18 = 0
33. The point to which the origin is to be shifted so that the point (3, 0) may change to (2, −3) is
A: (1, 3)
34. Matrix Matching
A: (i) q (ii) r (iii) p
A: (i) q (ii) s (iii) r (iv) p