JEE Mathematics - Coordinate Geometry - II Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
The equation of parabola whose focus is(-8,-2) and the directrix is 2x-y-9=0
- (a)
\(x^2-4xy+4y^2+116x+2y+259=0\)
- (b)
\(x^2+4xy+y^2+116x+2y+259=0\)
- (c)
\(x^2+4xy+4y^2-116x+2y+259=0\)
- (d)
\(x^2-4xy+4y^2+116x-2y+259=0\)
If y1,y2,y3 are the ordinates of the verticles of triangle inscribed in parabola y2=4ax, then area of the triangle, is
- (a)
\(\frac { 1 }{ 2a } |(y1-y)(y2-y3)(y3-y1)|\)
- (b)
\(\frac { 1 }{ 4a } |(y1-y)(y2-y3)(y3-y1)|\)
- (c)
\(\frac { 1 }{ 8a } |(y1-y)(y2-y3)(y3-y1)|\)
- (d)
\(\frac { 1 }{ 16a } |(y1-y)(y2-y3)(y3-y1)|\)
If the line x-1=0,is the directrix of the parabola y2-kx+8=0,then one of the value of k is
- (a)
\(1\over8\)
- (b)
8
- (c)
\(1\over4\)
- (d)
4
The coordinates of the points on the parabola y2=16x whose focal ditance is 8 units,are
- (a)
\((4,\pm8)\)
- (b)
\((2,\pm4\sqrt { 2 } )\)
- (c)
\((6,\pm4\sqrt { 6 } )\)
- (d)
None of these
The path of a projectile is a
- (a)
parabola
- (b)
ellipse
- (c)
hyperbola
- (d)
None of these
If (x1,y1),(x2,y2) and (x3,y3) be three points on the parabola y2 =4ax,the normals at which meet in a points then value of y1+y2 +y3 is
- (a)
0
- (b)
1
- (c)
-1
- (d)
None of these
Equation of the normal to the parabola y2 =4ax at the point (at2,2at),is
- (a)
ty=x+at2
- (b)
tx=y+at2
- (c)
y= -tx+2at+at3
- (d)
None of these
The point of intersection of the tangents at the points (at12 ,2at1) and (at22 ,2at2) is
- (a)
(at1t2,a(t1+t2))
- (b)
(a(t1+t2),at1t2)
- (c)
\(({a\sqrt{ t_1t_2},{a\over2}(t_1+t_2)})\)
- (d)
None of these
The normal at the point (at12,2at1) meets the parabola y2=4ax again at (at22,2at2); then
- (a)
\(t_2=t_1-{2\over t_1}\)
- (b)
\(t_2=-t_1-{2\over t_1}\)
- (c)
\(t_2=t_1+{2\over t_1}\)
- (d)
\(t_1t_2=2\)
The normals at the points (at12,2at1) and (at22,2at2) meet each other at the parabola again;then
- (a)
\(t_1t_2=-1\)
- (b)
\(t_1t_2=-2\)
- (c)
\(t_1t_2=1\)
- (d)
\(t_1t_2=2\)
If x+y =k is a normal to the parabola y2 =12x,then k equal
- (a)
3
- (b)
9
- (c)
-9
- (d)
-3
The equation of the common tangent touching the circle (x-3)2 + y2 =9 and the parabola y2 =4x above the x-axis,is
- (a)
\(\sqrt3y=3x+1\)
- (b)
\(\sqrt3y=-(x+3)\)
- (c)
\(\sqrt3.y=x+3\)
- (d)
\(\sqrt3y=-(3x+1)\)
The condition that the chord of the parabola y2 =4ax whose midpoint is (x1,y1) should subtend a right angle at the vertex, is
- (a)
y12 -2ax1+8a2=0
- (b)
y12 -ax1+8a2=0
- (c)
y12 -2ax1+7a2=0
- (d)
y12 -ax1+5a2=0
Three normals to the parabola y2 =x are drawn through a point (c,0);then
- (a)
\(c={1\over4}\)
- (b)
\(c={1\over2}\)
- (c)
\(c>{1\over2}\)
- (d)
None of these
The focus of the point from which two of the normals drawn to the parabola y2 =4ax are perpendicular to each other, is
- (a)
y2=x-3a
- (b)
y2=4a(x-3a)
- (c)
y2=a(x-3a)
- (d)
y2=a(3a-x)
The eccentricity of the ellipse 16x2+7y2=112 is
- (a)
\(4\over3\)
- (b)
\(7\over16\)
- (c)
\(3\over7\)
- (d)
\(3\over4\)
The length of the latus rectum of the ellipse 5x2+9y2=45 is
- (a)
\(5\over3\)
- (b)
\(10\over3\)
- (c)
\(2\sqrt5\over5\)
- (d)
\(\sqrt5\over3\)
The straight line y=4x+c is tangent to the ellipse \({x^{2}\over8}+{y^{2}\over4}=1\) Then c is equal to
- (a)
\(\pm4\)
- (b)
\(\pm6\)
- (c)
\(\pm1\)
- (d)
\(\pm132\)
The coordinates of the foci of the ellipse represented by the equation in example (49) are
- (a)
(3,\(\pm3\))
- (b)
(2,\(\pm2\))
- (c)
(1,\(\pm1\))
- (d)
None of these
The line x cos \(\alpha\) +y cos \(\alpha\) = p ,is tangent to the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) ; then value of a2 cos 2 \(\alpha\) + b2 sin2 \(\alpha\) is
- (a)
p
- (b)
p2
- (c)
\(1\over p^{2}\)
- (d)
None of these
The eccentricity of an ellipse whose pair of a conjugate diameters are y=x and 3y=-2x is
- (a)
\(2\over3\)
- (b)
\(1\over3\)
- (c)
\(1\over \sqrt3\)
- (d)
None of these
In the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\), the equation of the diameter conjugate to the diameter \(y={b\over a}{x}\) is
- (a)
\(y=-{b\over a}{x}\)
- (b)
\(y=-{a\over b}{x}\)
- (c)
\(x=-{b\over a}{y}\)
- (d)
none of these
If P(x,y), F1(3,0), F2(-3,0) and 16x2+25y2=400, then PF1+PF2 equals
- (a)
8
- (b)
6
- (c)
10
- (d)
12
The centre of the ellipse \({(x+y-2)^{2}\over 2}+{(x-y)^{2}\over16}=1\) is
- (a)
(0,0)
- (b)
(1,1)
- (c)
(0,1)
- (d)
(1,0)
The equation \({x^{2}\over 1-r}+{y^{2}\over 1+r}=1 \ (r>1)\) represents
- (a)
a circle
- (b)
a hyperbola
- (c)
an ellipse
- (d)
None of these
Let d be the perpendicular distance from the centre of the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) to the tangent drawn at point P on the ellipse. If F1 and F2 are two foci of the ellipse and (PF1-PF2)2=\(k({1}-{b^{2}\over d^{2}})\) then k=
- (a)
4a2
- (b)
3a2
- (c)
2a2
- (d)
None of these
Locus of the point of intersection of the perpendicular tangents to an ellipse is
- (a)
a straight line
- (b)
a circle
- (c)
an ellipse
- (d)
a hyperbola
Sum of the eccentric angles of the conormal points of the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) is
- (a)
an even number of \(\pi\)
- (b)
an odd number of \(\pi\)
- (c)
a prime number multiple of \(\pi\)
- (d)
None of these
The coordinates of the foci of the hyperbola 16x2-y2+64x+4y+44=0 is
- (a)
\((\pm \sqrt17-2.2)\)
- (b)
\((\pm \sqrt17+2.2)\)
- (c)
\((\pm \sqrt17-2)\)
- (d)
\((\pm \sqrt17-2)\)
The equation 9x2-16y2-18x-64y-199=0 represents
- (a)
a parabola
- (b)
an ellipse
- (c)
a hyperbola
- (d)
a pair of lines
The foci of the ellipse \({x^{2}\over 16^{2}}+{y^{2}\over b^{2}}=1\) and hyperbola \({x^{2}\over 144}+{y^{2}\over 81}={1\over25}\), coincide .Then value value b2 is
- (a)
1
- (b)
5
- (c)
7
- (d)
9
The equation of the hyperbola whose conjugate axis is 4 units ,tranverse axis lies along the x-axis and distance between the foci being 12 units is
- (a)
8x2-y2=32
- (b)
x2-8y2=32
- (c)
8y2-x2=32
- (d)
None of these
The equation of the hyperbola reffered to its axes as the axis of coordinates and whose distance between the two foci is 16 and ecentricity is \(\sqrt2\) is
- (a)
x2-y2=32
- (b)
y2-x2=32
- (c)
32x2-y2=1
- (d)
32y2-x2=1
If e and e' are the eccentricities of the hyperbolas \({x^{2}\over a^{2}}-{y^{2}\over b^{2}}=1\) and \({y^{2}\over b^{2}}-{x^{2}\over a^{2}}=1\) ,then value of (e)-2+(e')-2,is
- (a)
1
- (b)
2
- (c)
3
- (d)
None of these
From a focus of the hyperbola \({x^{2}\over a^{2}}-{y^{2}\over b^{2}}=1\), a perpendicular is drawn to a tangent on it. If M is the foot of perpendicular, then locus of M is
- (a)
x2+y2= a2+b2
- (b)
x2+y2=|a2-b2|
- (c)
x2+y2= a2
- (d)
None of these
If \({x^{2}\over cos^{2}\alpha}-{-y^{2}\over sin^{2}\alpha}=1\), represents family of hyperbolas,then which are constants
- (a)
Abscissae of foci
- (b)
Ordinates of foci
- (c)
both (a) and (b)
- (d)
None of these
The equation of the pair of lines passing through the origin and having slope \(m\epsilon 1\) for which equation (x - 3)(x + m) + 1 = 0 has integral roots is
- (a)
y2 - 6xy + 5x2 = 0
- (b)
y2 + 6xy - 5x2 = 0
- (c)
y2 + 6xy + 5x2 = 0
- (d)
y2 - 6xy - 5x2 = 0
If the lines represented by x2 - 2pxy - y2 = 0 are rotated about the origin through an angle \(\theta\) , one clockwise direction and other in anticlockwise direction, then the equation of the bisectors of the angle between the lines in the new position is
- (a)
px2 + 2xy - py2 = 0
- (b)
px2 + 2xy + py2 = 0
- (c)
x2 + 2pxy + y2 = 0
- (d)
none of these
If the sum of the slopes of the lines given by \(4{ x }^{ 2 }+2\lambda xy-7{ y }^{ 2 }=0\) is equal to the product of the slopes, then \(\lambda \) is equal to
- (a)
- 4
- (b)
4
- (c)
- 2
- (d)
2
Two of the straight lines given by 3x3 + 3x2y - 3xy2 + dy3 = 0 are at right angles, if
- (a)
d = - 1/3
- (b)
d = 1/3
- (c)
d = - 3
- (d)
d = 3
Mixed term xy is to be removed from the general equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, one should rotate the axes through an angle \(\theta\) then tan 2\(\theta\) equal to
- (a)
\(\frac { a-b }{ 2h } \)
- (b)
\(\frac { 2h }{ a+b } \)
- (c)
\(\frac { a+b }{ 2h } \)
- (d)
\(\frac { 2h }{ a-b } \)
Type of quadrilateral formed by the two pairs of lines 6x2 - 5xy - 6y2 = 0 and 6x2 - 5xy - 6y2 + x + 5y - 1 = 0 is
- (a)
square
- (b)
rhombus
- (c)
parallelogram
- (d)
rectangle
If the line y = mx is one of the bisector of the lines x2 + 4xy - y2 = 0, then the value of m is equal to
- (a)
\(\frac { -1+\sqrt { 5 } }{ 2 } \)
- (b)
\(\frac { 1+\sqrt { 5 } }{ 2 } \)
- (c)
\(\frac { -1-\sqrt { 5 } }{ 2 } \)
- (d)
\(\frac { 1-\sqrt { 5 } }{ 2 } \)
Two pairs of straight lines have the equations y2 + xy - 12x2 = 0 and ax2 + 2hxy + by2 = 0. One line will be common among them, if
- (a)
a = - 3(2h + 3b)
- (b)
a = 8(h - 2b)
- (c)
a = 2(b + h)
- (d)
a = - 3(b + h)
The equation ax2 + by2 + cx + cy = 0 represents a pair of straight lines, if
- (a)
a + b = 0
- (b)
c = 0
- (c)
a + c = 0
- (d)
c(a + b) = 0
If the angle between the lines x2 - xy + ay2 = 0 is 45°. then value(s) of a is/are
- (a)
- 6
- (b)
0
- (c)
6
- (d)
12
Equation of pair of lines passing through (1, -1) and parallel to the lines 2x2 + 5xy + 3y2 = 0 is
- (a)
2(x - 1)2 + 5(x - 1)(y + 1) + 3(y + 1)2 = 0
- (b)
3(x - 1)2 - 5(x - 1)(y + 1) + 2(y + 1)2 = 0
- (c)
2x2 + 5xy + 3y2 + x + y = 0
- (d)
3x2 - 5xy + 2y2 - 11x + 9y + 10 = 0
If the lines represented by 2x2 - 5xy + 2y2 = 0 be the two sides of a parallelogram and the line 5x + 2y = 1 be one of its diagonal.
On the basis of above information, answer the following questions:
The equation of the other diagonal is
- (a)
10x - 11y = 0
- (b)
11x - 10y = 0
- (c)
3x - 2y = 0
- (d)
2x - 3y = 0
Let \({ f }_{ 1 }\left( x,y \right) \equiv { ax }^{ 2 }+2hxy+b{ y }^{ 2 }=0\) and let \({ f }_{ i+1 }\left( x,y \right) =0\) denotes the equation of the bisectors of \({ f }_{ i }\left( x,y \right) =0\) for all i = 1, 2, 3, ....
On the basis of above information, answer the following questions:
If fi+1(x, y) = 0 represents the equation of a pair of perpendicular lines, then f3 (x, y) = 0 is
- (a)
bx2 - 2hxy - ay2 = 0
- (b)
ax2 + 2hxy + by2 = 0
- (c)
ax2 - 2hxy + by2 = 0
- (d)
bx2 - 2hxy + ay2 = 0
Let \({ f }_{ 1 }\left( x,y \right) \equiv { ax }^{ 2 }+2hxy+b{ y }^{ 2 }=0\) and let \({ f }_{ i+1 }\left( x,y \right) =0\) denotes the equation of the bisectors of \({ f }_{ i }\left( x,y \right) =0\) for all i = 1, 2, 3, ....
On the basis of above information, answer the following questions:
If fi+1(x, y) = 0 represents the equation of a pair of perpendicular lines, then \({ f }_{ n+2 }\left( x,y \right) =0\forall n\ge 2\) is same as
- (a)
\({ f }_{ n+2 }\left( x,y \right) =0\)
- (b)
\({ f }_{ n+1 }\left( x,y \right) =0\)
- (c)
\({ f }_{ n }\left( x,y \right) =0\)
- (d)
none of the above