Mathematics - Coordinate Geometry I 1
Exam Duration: 45 Mins Total Questions : 30
The orthocentre of the triangle whose vertices are A(0, 0), B(3, 4), C(4, 0) is
- (a)
\((3, \frac {4}{5})\)
- (b)
\((3, \frac {5}{6})\)
- (c)
\((3, \frac {6}{7})\)
- (d)
\((3, \frac {3}{4})\)
If x1, x2, x3 are in G.P with common ration and y1, y2, y3 are alos in G.P with common ratio s, then area of the \(\triangle\)ABC, where A(x1, y1), B(x2, y2), C(x3, y3) is
- (a)
\(\frac 1{2} |x_1y{_1}(1-r)(1-s)(r-s)|\)
- (b)
\(\frac 1{2} |x_1y{_1}(1-r)(1-s)|\)
- (c)
\(\frac 1{2} |x_1y{_1}(1-r)(r-s)|\)
- (d)
\(\frac 1{2} |x_1y{_1}(1-s)(r-s)|\)
If x1, x2, x3 and y1, y2, y3 are in G.p with same common ratio, then the points (x1, y1), (x2, y2), (x3, y3) are
- (a)
vertices of an isosceles triangle
- (b)
vertices of an equilateral triangle
- (c)
vertices of a right angled triangle
- (d)
collinear
Number of points with integral coordinates that lie inside the triangle whose vertices are (0, 0), (0, 21), (21, 0) is
- (a)
180
- (b)
190
- (c)
200
- (d)
210
The equation fo the line passing through the point (4, 5) and making an angle of \(\frac {\pi} {4}\) with the line 2x-y+7=0 is
- (a)
3x-y+17=0
- (b)
3x+y=17
- (c)
x-3y+11=0
- (d)
None of these
The points (1, 2) and (3, 4) are on the same side of the straight line 3x-5y+k=0, then
- (a)
7
- (b)
k=7
- (c)
k<7 or k>11
- (d)
k=11
The equation of the straight line passing through the point of intersection of the lines 5x-3y=1 and 2x+3y=23 and perpendicular to the line 5x-3y=1, is
- (a)
63x+105y=781
- (b)
105x+63y=781
- (c)
63x-105y=781
- (d)
105x-63y=781
The area bounded by the lines
\(y=\left| x \right| -1\)
\(y=-\left| x \right| -1\), is
- (a)
1
- (b)
2
- (c)
\(2\sqrt { 2 } \)
- (d)
4
The angle between the lines represented by ax2+2hxy+by2=0 is
- (a)
\(\tan ^{ -1 }{ \left( \cfrac { 2\sqrt { { h }^{ 2 }-ab } }{ a+b } \right) } \)
- (b)
\(\tan ^{ -1 }{ \left( \cfrac { 2\sqrt { { ab }- } { h }^{ 2 } }{ a+b } \right) } \)
- (c)
\(\tan ^{ -1 }{ \left( \cfrac { 2\sqrt { { { h }^{ 2 } }+ } ab }{ a-b } \right) } \)
- (d)
NONE OF THESE
The lines given by ax2+2hxy-bx2=0 bisect the angle between the coordinate axes; then the value of (a+b)2 is
- (a)
h2
- (b)
2h2
- (c)
3h2
- (d)
4h2
The equation ax2+2hxy2+by2+2gx+2fy+c=0 represents a pair of straight lines if
- (a)
2abc+fgh-af2-bg2-ch2=0
- (b)
abc+2fgh-af2-bg2-ch2=0
- (c)
af2+bg2+ch2-2fgh-abc=0
- (d)
NONE OF THESE
The point of intersection ofthe lines given by ax2+2hxy2+by2+2gx+2fy+c=0 , is
- (a)
\(\left( \cfrac { hf+bg }{ ab-{ h }^{ 2 } } ,\cfrac { gh-af }{ ab-{ h }^{ 2 } } \right) \)
- (b)
\(\left( \cfrac { hf+bg }{ ab+{ h }^{ 2 } } ,\cfrac { gh+af }{ ab+{ h }^{ 2 } } \right) \)
- (c)
\(\left( \cfrac { hf+bg }{ ab-{ h }^{ 2 } } ,\cfrac { gh+af }{ ab-{ h }^{ 2 } } \right) \)
- (d)
\(\left( \cfrac { hf-bg }{ ab-{ h }^{ 2 } } ,\cfrac { gh-af }{ ab-{ h }^{ 2 } } \right) \)
The product of the lengths ofthe perpendiculars drawn from a point (x1, y1) to the lines represented by ax2+2hxy+by2=0, is
- (a)
\(\cfrac { a{ x }_{ 1 }^{ 2 }+2h{ x }_{ 1 }{ y }_{ 1 }+b{ y }_{ 1 }^{ 2 } }{ \sqrt { { (a+b) }^{ 2 }+4{ h }^{ 2 } } } \)
- (b)
\(\cfrac { a{ x }_{ 1 }^{ 2 }+2h{ x }_{ 1 }{ y }_{ 1 }+b{ y }_{ 1 }^{ 2 } }{ \sqrt { { (a-b) }^{ 2 }+4{ h }^{ 2 } } } \)
- (c)
\(\cfrac { a{ x }_{ 1 }^{ 2 }+2h{ x }_{ 1 }{ y }_{ 1 }+b{ y }_{ 1 }^{ 2 } }{ \sqrt { { 4{ h }^{ 2 }-(a+b) }^{ 2 } } } \)
- (d)
\(\cfrac { a{ x }_{ 1 }^{ 2 }+2h{ x }_{ 1 }{ y }_{ 1 }+b{ y }_{ 1 }^{ 2 } }{ \sqrt { { 4{ h }^{ 2 }-(a-b) }^{ 2 } } } \)
A square is inscribed in the circle x2+y2-2x+4y+3=0. Its sides are parallel to the coordinate axes. Then one vertex of the square is
- (a)
\((1+\sqrt { 2 } ,-2)\)
- (b)
\((1-\sqrt { 2 } ,-2)\)
- (c)
\((1,-2+\sqrt { 2 } )\)
- (d)
NONE OF THESE
The equation of the tangents to the circle x2+y2=25 which are inclined at angle of \({ 30 }^{ \circ }\) to the x-axis are
- (a)
\(\pm \sqrt { 3 } y=x+10\)
- (b)
\(\sqrt { 3 } y=x\pm 10\)
- (c)
\(y=x\sqrt { 3 } \pm 5\)
- (d)
NONE OF THESE
If the circles x2+y2+2x+2ky+6=0 and x2+y2+2ky+k=0, intersects ortogonally, then k is
- (a)
\(2or\frac { -3 }{ 2 } \)
- (b)
\(-2or\frac { -3 }{ 2 } \)
- (c)
\(2or\frac { 3 }{ 2 } \)
- (d)
\(-2or\frac { 3 }{ 2 } \)
If a>2b>0, then the positive values of m for which \(y=mx-b\sqrt { 1+{ m }^{ 2 } } \), is a common tangent to x2+y2=b2 and (x-a)2+y2=b2, is
- (a)
\(\frac { 2b }{ \sqrt { { a }^{ 2 }-{ 4b }^{ 2 } } } \)
- (b)
\(\frac { \sqrt { { a }^{ 2 }-{ 4b }^{ 2 } } }{ 2b } \)
- (c)
\(\frac { 2b }{ a-2b } \)
- (d)
\(\frac { b }{ a-2b } \)
The number of integral values of m for which the x coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is
- (a)
2
- (b)
0
- (c)
4
- (d)
1
All the points lying inside the triangle formed by the points (0,4), (2, 5) and (6, 2) satisfy
- (a)
3x + 2y + 8~ 0
- (b)
2x + y -10~ 0
- (c)
2x - 3y - 11 ~ 0
- (d)
- 2x + Y - 3 ~ 0
The graph of the function y = cos x cos (x + 2) -- cos2(x + 1) is
- (a)
a straight line passing through (0, - sin2 1) with slope 2
- (b)
a straight line passing through (0, 0)
- (c)
a parabola with vertex (1, - sin21)
- (d)
a ~traight line passing through the point \(\left( \frac { \pi }{ 2 } -sin^{ 2 }1 \right) \) are parallel to the x -axis
If two vertices of an equilateral triangle have integral coordinates, then the third vertex will have
- (a)
integral coordinates
- (b)
coordinates which are rational
- (c)
at least one coordinate irrational
- (d)
coordinates which are irrational
If P \((1+\alpha /.\sqrt { 2 } ,2+\alpha /\sqrt { 2 } )\) be any point on a line, then the range of values of t for which the point P lies between the parallel lines x + 2y = 1 and 2x + 4y = 15 is
- (a)
\(-\frac { 4\sqrt { 2 } }{ 3 } <\alpha <\frac { 5\sqrt { 2 } }{ 6 } \qquad \)
- (b)
\(0<\alpha <\frac { 5\sqrt { 2 } }{ 6 } \)
- (c)
\(-\frac { 4\sqrt { 2 } }{ 3 } <\alpha <0\)
- (d)
none of these
The locus of a point which moves such that the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distances from the other two sides is
- (a)
a pair of straight lines
- (b)
a parabola
- (c)
an ellipse
- (d)
a hyperbola
Two points A and B move on the x-axis and the y-axis respectively such that the distance between the two points is always-the same. The locus of the middle point of AB is
- (a)
a straight line
- (b)
a circle
- (c)
a parabola
- (d)
an ellipse
The set of values of 'b' for which the origin and the point (1,1) lie on the same side of the straight line a2x + aby + 1 = 0, \(\forall a\in R,b>0\) are
- (a)
\(b\in (2,4)\)
- (b)
\(b\in (0,2)\)
- (c)
\(b\in [0,2]\)
- (d)
none of these
The equations (b - c) x + (c - a) y + a - b = 0 (b3 _ c3) X + (c3 - a3) y + a3 - b3 = 0 will represent the same line if
- (a)
b=c
- (b)
c=a
- (c)
a=b
- (d)
a+b+c=0
Equation of a straight line passing through the point of intersection of x - y + 1 = 0 and 3x + y - 5 = 0 are perpendicular to one of them is
- (a)
x + y + 3 = 0
- (b)
x + y - 3 = 0
- (c)
x- 3y - 5 = 0
- (d)
- 3y + 5 = 0
Let L be the line 2x + y = 2. If the axes are rotated by 45°, then the intercept made by the line L on the length of new axes are respectively
- (a)
\(\sqrt { 2 } \) and 1
- (b)
1 and \(\sqrt { 2 } \)
- (c)
\(\sqrt [ 2 ]{ 2 } and\quad \sqrt [ 2 ]{ 2/3 } \)
- (d)
\(\sqrt [ 2 ]{ 2 } /3and\quad \sqrt [ 2 ]{ 2 } \)
The coordinates of the points A and B an' respectively (-3,2) and (2, 3). P and Q are points on the line joining A and B such that AP == PO = QB. A square PQRS is constructed on PQ as one side, the coordinates of R can be
- (a)
\(\left( -\frac { 4 }{ 3 } ,\frac { 7 }{ 3 } \right) \)
- (b)
\(\left( 0,\frac { 13 }{ 3 } \right) \)
- (c)
\(\left( \frac { 1 }{ 3 } ,\frac { 8 }{ 3 } \right) \)
- (d)
\(\left( \frac { 2 }{ 3 } ,1 \right) \)
Locus of centroid of the triangle whose vertices are a cost, a sin r ), (b sint, - b cost) and (1,0), where t is a parameter, is
- (a)
(3x - 1)2 + (3y)2 = a2 - b2
- (b)
(3x - 1)2 + (3y)2 =-a2 + b2
- (c)
{3x + ])2 + (3y)2 =- a2 + b2
- (d)
(3x+1)2+(3y)2=a2_b