Mathematics - Coordinate Geometry II
Exam Duration: 45 Mins Total Questions : 30
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)
\(\frac { 1 }{ 16a } |(y1-y)(y2-y3)(y3-y1)|\)
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
The equation of the directrix of the parabola \(4y^2=+12x-12y+39=0\) is
- (a)
\(4x-7=0\)
- (b)
\(7x-4=0\)
- (c)
\(4x+7=0\)
- (d)
\(7x+4=0\)
The line \(lx+my+n=0 \) touches the parabola \(y^2=4ax\) if
- (a)
\(lm=an^2\)
- (b)
\(ln=an^2\)
- (c)
\(mn=al^2\)
- (d)
None of these
The point of intersection of the tangents at the ends of the latus rectum of parabola y2=4x,is
- (a)
(-1,-1)
- (b)
(0,-1)
- (c)
(-1,0)
- (d)
(1,1)
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
If the segment of the line lx+my+n =0 intercepted by the parabola y2 =4ax, subtends a right angle at the vertex ,then
- (a)
al+n=0
- (b)
4am+n=0
- (c)
4al+4am+n=0
- (d)
4al+n=0
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 length of the latus rectum of the ellipse 5x2+9y2=45 is
- (a)
\(5\over3\)
- (b)
\(10\over3\)
- (c)
\(2\sqrt5\over5\)
- (d)
\(\sqrt5\over3\)
If eccentricity of ellipse becomes zero ,then it takes the form of
- (a)
a circle
- (b)
a parabola
- (c)
a straight line
- (d)
None of these
The locus of the midpoints of the focal chord of the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) is
- (a)
\({x^{2}\over a^{2}}+{y^{2}\over b^{2}}={ex\over a}\)
- (b)
\({x^{2}\over a^{2}}-{y^{2}\over b^{2}}={ex \over a}\)
- (c)
x2+y2=a2
- (d)
x2+y2=a2 + b2
The number of normals that can be drawn to a hyperbola from a goven point is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
The locus of the point of intersection of perpendicular tangents to the hyperbola \({x^{2}\over a^{2}}-{y^{2}\over b^{2}}=1\) is
- (a)
x2+y2=|a2-b2|
- (b)
x2+y2= a2
- (c)
x2+y2= b2
- (d)
x2+y2= a2+b2
The locus of the midpoints of the chords of the circle x2+y2=16 which are tangents to the hyperbola 9x2-16y2=144 is
- (a)
(x2+y2)2=16x2-9y2
- (b)
(x2+y2)2=9x2-16y2
- (c)
(x2+y2)2=16x2+9y2
- (d)
None of these
The locus of the centre of the circle which touches two given circles externally,is
- (a)
a parabola
- (b)
an ellipse
- (c)
a hyperbola
- (d)
None of these
If the two pairs of lines x2 - 2mxy - y2 = 0 and x2 - 2nxy - y2 = 0 are such that one of them represents the bisectors of the angles between the other, then
- (a)
mn + 1 = 0
- (b)
mn - 1 = 0
- (c)
1/m + 1/n = 0
- (d)
1/m - 1/n = 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
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
Equation of pair of straight lines drawn through (1,1) and perpendicular to the pair of lines 3x2 - 7xy - 2y2 = 0 is
- (a)
2x2 + 7xy - 11x + 6 = 0
- (b)
2(x - 1)2 + 7(x - 1)(y - 1) - 3y2 = 0
- (c)
2(x - 1)2 + 7(x - 1)(y - 1) - 3(y - 1)2 = 0
- (d)
none of the above
The equation of image of pair of lines \(y=\left| x-1 \right| \) in y-axis is
- (a)
x2 + y2 + 2x + 1 = 0
- (b)
x2 - y2 + 2x - 1 = 0
- (c)
x2 - y2 + 2x + 1 = 0
- (d)
none of these
If the two lines represented by \({ x }^{ 2 }\left( \tan ^{ 2 }{ \theta } +\cos ^{ 2 }{ \theta } \right) -2xy\tan { \theta } +{ y }^{ 2 }\sin ^{ 2 }{ \theta } =0\) make angles \(\alpha ,\beta \) with the x-axis, then
- (a)
\(\tan { \alpha } +\tan { \beta } =4cosec2\theta \)
- (b)
\(\tan { \alpha } \tan { \beta } =\sec ^{ 2 }{ \theta } +\tan ^{ 2 }{ \theta } \)
- (c)
\(\tan { \alpha } -\tan { \beta } =2\)
- (d)
\(\frac { \tan { \alpha } }{ \tan { \beta } } =\frac { 2+\sin { 2\theta } }{ 2-\sin { 2\theta } } \)
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
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