JEE Mathematics - Coordinate Geometry - I Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
The points (-a, -b), (0, 0), (a, b) and (a2, ab) are
- (a)
collinear
- (b)
vertices of a parallelogram
- (c)
vertices of a rectangle
- (d)
None of these
If the vertices of a triangle have integral coordinates, then the triangle is
- (a)
equilateral
- (b)
never equilateral
- (c)
always isosceles
- (d)
None of these
An equilateral triangle has one vertex at the point (3, 4) and another at the point (-2, 3). The co-ordinates of the third vertex are
- (a)
(1,1) or (1, -1)
- (b)
\((\frac { 1+\sqrt { 3 } }{ 2 } ,\frac { 7-5\sqrt { 3 } }{ 2 } )\quad or\quad (\frac { 1-\sqrt { 3 } }{ 2 } ,\frac { 7+5\sqrt { 3 } }{ 2 } )\)
- (c)
\((-\sqrt { 3 } ,\sqrt { 3 } )\quad or\quad (\sqrt { 3 } ,\sqrt { 3 } )\)
- (d)
None of these
If the vertices P,Q,R are rational points, which of the following points of the triangel PQR is (are) always rational points?
- (a)
centroid
- (b)
incentre
- (c)
circumcentre
- (d)
orthocentre
Let 0<\(\alpha\)<\(\frac { \pi }{ 2 } \) be a fixed angle. If P(cos\(\theta \), sin\(\theta \)) and Q (cos(\(\alpha - \theta\)), sin(\(\alpha - \theta\))), then Q is obtained from P by
- (a)
clock wise raotaion around the origin through an angle \(\alpha\)
- (b)
anticlockwise rotation around the origin
- (c)
reflection in the line through the origin with sope tan \(\alpha\)
- (d)
reflection in the line through the origin with slope tan \((\frac {\alpha}{2})\)
Area of the triangle formed by the llines y=m1x+c1, y=m2x+c2, x=0 is
- (a)
\(\begin{vmatrix} \frac { { c }_{ 2 }-{ c }_{ 1 } }{ ({ m }_{ 2 }-{ m }_{ 2 })^2 } \end{vmatrix}\)
- (b)
\(\frac { 1 }{ 2 } \begin{vmatrix} \frac { ({ c }_{ 2 }-{ c }_{ 1 })^{ 2 } }{ ({ m }_{ 2 }-{ m }_{ 2 }) } \end{vmatrix}\)
- (c)
\(\frac { 1 }{ |m_2-m_1|^2 } \)
- (d)
None of these
The coordinates of the foot of perpendicular from (2, 3) on the line 2x+3y+5=0, are
- (a)
\((-\frac {16}{13}, \frac {24}{13})\)
- (b)
\((\frac {-16}{13}, \frac {-24}{13})\)
- (c)
\((\frac {16}{13}, \frac {-24}{13})\)
- (d)
\((\frac {16}{13}, \frac {24}{13})\)
The algebraic sum of the perpendicular distances of the points (1,0), (0,1), (2,2) from a variable line is zero; then the line passes through a fixed point
- (a)
(1, 1)
- (b)
(2, 3)
- (c)
(2, 4)
- (d)
(4, 3)
The coordinates of the four vertices of a quaddrilateral taken in order are (-2, 4), (-1, 2), (1, 2) and (2, 4). The equation of the line passing through the point (-1,2) and dividing the quadrilateral into two parts of equal areas is
- (a)
x+1=0
- (b)
x+y=1
- (c)
x-y+3=0
- (d)
NONE OF THESE
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
If pair of lines x2-2pxy-y2=0 and x2-2qxy-y2=0 be such that pair bisects the angle between the other pair, then value of pq is
- (a)
-1
- (b)
1
- (c)
-2
- (d)
2
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 angle between the lines represented by ax2+2hxy+by2=0 and ax2+2hxy+by2+2gx+2fy+c=0 are
- (a)
equal
- (b)
complementary
- (c)
supplementary
- (d)
NONE OF THESE
The equation ax2+2hxy+by2+2gx+2fy+c=0 represents a circle if
- (a)
a = b, h = 0
- (b)
a = b, c = 0
- (c)
g = f, c = 0
- (d)
NONE OF THESE
The lines joining the origin to the point of intersection of the curves
ax2+2hxy+by2+2gx=0,
a'x2+2h'xy+b'y2+2g'x=0,
are perpendicular to each other, if
- (a)
(a-b)g' = (a'-b')g
- (b)
(a+b)g = (a'+b')g
- (c)
(a+b)g' = (a'+b')g
- (d)
NONE OF THESE
The equation of the pair of straight lines joining the points of intersection of the line y = 3x+2 and the curve x2+2xy-3y2+4x+8y-11=0 with origin, is
- (a)
7x2+2xy-y2=0
- (b)
7x2-2xy+y2=0
- (c)
7x2-2xy-y2=0
- (d)
NONE OF THESE
The equation of the circle which cuts an intercept of 8 units on the x-axis and touches the y-axis at(0, 3)
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }-10x\pm 6y+9=0\)
- (b)
\({ x }^{ 2 }+{ y }^{ 2 }+10x\pm 6y+9=0\)
- (c)
\({ x }^{ 2 }+{ y }^{ 2 }\pm 10x-6y+9=0\\ \)
- (d)
NONE OF THESE
The circle x2+y2-2ax-2ay+a2=0,
- (a)
touches the x-axis only
- (b)
touches the y-axis only
- (c)
touches both the axes
- (d)
NONE OF THESE
The absissae of the end points of a diameter of circle are the roots of the equation x2+2 px-q2=0 and the coordinates of these points are the roots of the equation x2+2rx-s2=0; then radius of the circle is
- (a)
\(\sqrt { { p }^{ 2 }+{ q }^{ 2 } } \)
- (b)
\(\sqrt { { r }^{ 2 }+{ s }^{ 2 } } \)
- (c)
\(\sqrt { { p }^{ 2 }+{ q }^{ 2 }-{ r }^{ 2 }-{ s }^{ 2 } } \)
- (d)
\(\sqrt { { p }^{ 2 }+{ q }^{ 2 }+{ r }^{ 2 }+{ s }^{ 2 } } \)
If the two circles x2+y2=r2 and (x-5)2+y2=9 intersect in two distinct points,then
- (a)
2
- (b)
r<2
- (c)
r=2
- (d)
r>2
The area of the triangle formed by the tangent and the chord of contact from the point (3,4) to the circle x2+y2=9, is
- (a)
\(\frac { 32 }{ 25 } \)
- (b)
\(\frac { 64 }{ 25 } \)
- (c)
\(\frac { 72 }{ 25 } \)
- (d)
\(\frac { 192 }{ 25 } \)
The equation of the circle passing through the point (1, 1) and points of intersection of the circles x2+y2=6 and x2+y2-6x+8=0 is given by
- (a)
x2+y2 -6x+9=0
- (b)
x2+y2 -3x+1=0
- (c)
x2+y2 -4y+2=0
- (d)
NONE OF THESE
The length of the tangent from the point (5, 1) to the circle x2+y2+6x-4y-3=0 is
- (a)
81
- (b)
29
- (c)
7
- (d)
NONE OF THESE
If two distinct chords, drawn from the point (p,q) on the circle x2+y2=px+qy (where pq\(\neq \) 0) are bisected by the x-axis, then
- (a)
p2=q2
- (b)
p2=8q2
- (c)
p2<8q2
- (d)
p2>8q2
Equation of a diameter of the circle x2+y2-6x-2y=0 is
- (a)
x-3y=0
- (b)
x+3y=0
- (c)
3x-y=0
- (d)
NONE OF THESE
The angle of intersection of two circles x2+y2+8x-2y-9=0 and x2+y2-2x+8y-7=0 is
- (a)
\({ 30 }^{ \circ }\)
- (b)
\({ 45 }^{ \circ }\)
- (c)
\({ 60 }^{ \circ }\)
- (d)
\({ 90 }^{ \circ }\)
The number of the common tangent to the circle x2+y2=4 and x2+y2-6x-8y=24, is
- (a)
0
- (b)
1
- (c)
3
- (d)
4
The vertices of a triangle are A \((x_{ 1 },x_{ 1 }tan\alpha )\) B \((x_{ 2 },x_{ 3 }tan\gamma )\) If the circumstance of ABC coincides with the origin and H (a, b) be its ortocentre then \(\frac { a }{ b } \) is equal to
- (a)
\(\frac { cos\alpha +cos\beta +cos\gamma }{ cos\alpha cos\beta \quad cos\gamma } \)
- (b)
\(\frac { sin\alpha +sin\beta +cos\gamma }{ sin\alpha \quad sin\beta \quad sin\gamma } \)
- (c)
\(\frac { tan\alpha +tan\beta +tan\gamma }{ tan\alpha \quad tan\beta \quad tan\gamma } \)
- (d)
\(\frac { cos\alpha +cos\beta +cos\gamma }{ sin\alpha \quad sin\beta \quad sin\gamma } \)
The image of P (a, b) on y = -' X is Q and the image of Q on the line y = X is R. Then the mid point of R is
- (a)
(a + b, b + a)
- (b)
\(\left( \frac { a+b }{ 2 } ,\frac { b+a }{ 2 } \right) \)
- (c)
(a-b,b+a)
- (d)
(0,0)
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
Let n be the number of points having rational coordinates equidistant from the point (0,13), then
- (a)
\(n\le 1\)
- (b)
n = 1
- (c)
\(n\le 2\)
- (d)
n>2
If \(p_{ 1 },p_{ 2 },p_{ 3 }\) the length perpendiculars from the points \((m_{ 2 },2m)(m^{ ' },m+m^{ ' })\) and \((m^{ 2 },2m^{ ' })\) respectively on the line
\(xcos\alpha +ysin\alpha +\frac { sin^{ 2 }\alpha }{ cos\alpha } =0,p_{ 1 },p_{ 2 },p_{ 3 }\)
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
If the point (a, a) fall between the lines | x + y | = 2, then
- (a)
|a|=2
- (b)
|a|=1
- (c)
|a|<1
- (d)
\(|a|<\frac { 1 }{ 2 } \)
The diagonals of the parallelogram whose sides are Ix + my + n = 0, Ix + my + n' = 0, mx + ly + n = 0, mx + ly + n' = °include an angle
- (a)
\(\pi /3\)
- (b)
\(\pi /2\)
- (c)
\(tan^{ -1 }\left( \frac { l^{ 2 }-m^{ 2 } }{ l^{ 2 }+m^{ 2 } } \right) \)
- (d)
\(tan^{ -1 }\left( \frac { 2lm }{ l^{ 2 }+m^{ 2 } } \right) \)
The position of a moving point in the xy-plane at time t is given by \(\left( ucos\alpha \quad t,\quad u\quad sin\quad \alpha t-\frac { 1 }{ 2 } gt^{ 2 } \right) \)where u,a, g are constants. The locus of the moving point is
- (a)
a circle
- (b)
a parabola
- (c)
an ellipse
- (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
If (- 6, - 4), (3, 5), (-2, 1) are the vertices of a parallelogram, then remaining vertex cannot be
- (a)
(0,-1)
- (b)
(-1,0)
- (c)
(-11, -8)
- (d)
(7, 10)
If each of the points (x,4), (- 2, YI) lies on the line joining the points (2, - 1), (5, - 3), then the point P (x1' Y1) lies on the line
- (a)
6(x+'y)-25=0
- (b)
2x+6y+1=0
- (c)
2x + 3y - 6 = 0
- (d)
6 (x + y) - 23 = 0
If the area of the triangle whose vertices are (b, C), (C, a) and (a, b) is ~, then the area of triangle whose vertices are and (a, b) is \(\triangle \) then the area of triangle whose vertices are \((ac-b^{ 2 },ab-c^{ 2 })\quad (ba-c^{ 2 },bc-a^{ 2 })\) and \((cb-a^{ 2 },ca-b^{ 2 })\)
- (a)
\(\triangle ^{ 2 }\)
- (b)
\((a+b+c)^{ 2 }\triangle \)
- (c)
\(a\triangle +b\triangle ^{ 2 }\)
- (d)
\(2^{ n }+3\)
If P (1, 0), Q (-1, 0) and R (2, 0) are three given points, then the locus of points satisfying the relation \((SQ)^{ 2 }+(SR)^{ 2 }=2(SP)^{ 2 }\)
- (a)
a straight line parallel to x-axis
- (b)
circle through origin
- (c)
circle with centre at the origin
- (d)
a straight line parallel to y-axis
A family of lines is given by the equation (3x + 4y + 6) + (x + y + 2) = O. The line situated at the greatest distance from the point (2, 3) belonging to this family has the equation
- (a)
15x + 8y + 30 = 0
- (b)
4x + 3y + 8 = 0
- (c)
5x + 3y + 6 = 0
- (d)
5x + 3y + 10 = 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
If bx + cy = a, where a, b, c are the same sign, be a line such that the area enclosed by the line and the axes of reference is \(\frac { 1 }{ 8 } unit^{ 2 }\) then
- (a)
b, a, c are in GP
- (b)
b, 2a, c are in GP
- (c)
\(b,\frac { a }{ 2 } ,c\) are in AP
- (d)
b, - 2a, c are in GP
Let \(L_{ 1 }\) = ax + by + a \(\sqrt [ 3 ]{ b } =0 \) and L2 == bx - ay + b \(\sqrt [ 3 ]{ a } =0\) be two straight lines. The equations of the bisectors of the angle formed by the loci. Whose equations are \(\lambda _{ 1 }L_{ 1 }-\lambda _{ 2 }L_{ 2 }=0\) and \(\lambda _{ 1 }L_{ 2 }-\lambda _{ 2 }L_{ 2 }=0\) \(\lambda _{ 1 } \) and \(\lambda _{ 2 }\) being non zero real numbers, are given by
- (a)
\(L_{ 1 }=0\)
- (b)
\(L_{ 2 }=0\)
- (c)
\(\lambda _{ 1 }L_{ 1 }-\lambda _{ 2 }L_{ 2 }=0\)
- (d)
\(\lambda _{ 1 }L_{ 2 }-\lambda _{ 2 }L_{ 2 }=0\)
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)
\(\sqrt [ 2 ]{ 2 } /3and\quad \sqrt [ 2 ]{ 2 } \)
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
For points P == (x1, y1) and Q = (x2, y2) of the coordinate plane, a new distance d (P, Q) is defined by d(P, Q) = |x1 - x2| + |y1 - y2| Let 0 (0, 0), A = (1, 2), B = (2,3) and C \(\equiv \) (4, 3) are four fixed points on X - Y plane
Nine point centre of the \(\triangle\)ABC is
- (a)
(1, 6)
- (b)
(1, 5/2)
- (c)
(3, 1)
- (d)
(2, 7/2)
If the equation of the locus of a point equidistant from the points \((a_{ 1 },b_{ 1 })\) and \((a_{ 2 },b_{ 2 })\)
(a1-a2)x+(bi-b2)y+r=-0
- (a)
\(\frac { 1 }{ 2 } ({ a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 }-{ a }_{ 2 }^{ 1 }-{ b }_{ 1 }^{ 2 })\)
- (b)
\(({ a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 }-{ a }_{ 2 }^{ 1 }-{ b }_{ 1 }^{ 2 })\qquad \)
- (c)
\(\frac { 1 }{ 2 } ({ a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 }-{ a }_{ 2 }^{ 1 }-{ b }_{ 1 }^{ 2 })\qquad \)
- (d)
\(({ a }_{ 2 }^{ 2 }+{ b }_{ 2 }^{ 2 }-{ a }_{ 2 }^{ 1 }-{ b }_{ 1 }^{ 2 })\qquad \)
The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P (h, k) with the Jines y = x and x + y =-2is 4h2. Then the locus of the point P is
- (a)
y=3x-2 or y=-3x-2
- (b)
y=2x-1 or y=-3x-2
- (c)
y=3x+2 or y=-3x+2
- (d)
y=2x+1 or y=-2x+1