JEE Main Mathematics - Coordinate Geometry II
Exam Duration: 60 Mins Total Questions : 30
The coordinates of the vertex and focus of the parabola \(y^2-8y-x+19=0 \)are respectively
- (a)
\(_{ }(3,4),\left( \frac { 13 }{ 4 } ,4 \right) \)
- (b)
\(_{ }(-3,-4),\left( \frac { 13 }{ 4 } ,4 \right) \)
- (c)
\(_{ }(3,4),\left( \frac { -13 }{ 4 } ,-4 \right) \)
- (d)
None of these
If the vertex of a parabola is(-5,0) and directrix of the line x+3=0 then equation of the parabola,is
- (a)
y2=4x+16
- (b)
y2=-4x+16
- (c)
y2=4x-16
- (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 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 the parabola y2 =4ax,passes through the point (1,2) then length of the latus rectum is
- (a)
4
- (b)
3
- (c)
2
- (d)
1
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 locus of the point from which two of the normals drawn to the parabola y2= 4ax are coincident, is
- (a)
27ay2 =(2a-x)3
- (b)
27ay2 =2(2a-x)3
- (c)
27ay2 =4(2a-x)3
- (d)
None of these
A point moves so that sum of its distance from two fixed points is always constant ,then the locus of this moving point is a
- (a)
straight line
- (b)
circle
- (c)
parabola
- (d)
ellipse
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 straight line lx+my+n=0 is normal to the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) if
- (a)
\({a^{2}\over m^{2}}+{b^{2}\over l^{2}}=({a^{2}-b^{2}\over n^{2}})\)
- (b)
\({a^{2}\over l^{2}}+{b^{2}\over m^{2}}={(a^{2}-b^{2})\over n^{2}}\)
- (c)
\({a^{2}\over l^{}}-{b^{2}\over m^{2}}={(a^{2}-b^{2})^{2}\over n^{2}}\)
- (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
Let E be the ellipse \({x^{2}\over 9}+{y^{2}\over 4}=1\) and c be the circle x2+y2=9 .Let P and Q be points (1,2) and (2,1) respectively.Then
- (a)
Q lies inside C but outside E
- (b)
Q lies outside both C and E
- (c)
P lies both inside C and E
- (d)
P lies inside C but outside E
A tangent to the ellipse x2+4y2=4 meets the ellipse x2+2y2=6 at P and Q .The angle between the tangents at P and Q of the ellipse x2+2y2=6 ,is
- (a)
\(\pi\over2\)
- (b)
\(\pi\over3\)
- (c)
\(\pi\over4\)
- (d)
\(\pi\over6\)
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 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 point of contact of the tangent y=x+2 hyperbola 5x2-9y2=45 is
- (a)
\(({{9\over2},{5\over2}})\)
- (b)
\(({{5\over2}},{{9\over2}})\)
- (c)
\(({{-9\over2},{-5\over2}})\)
- (d)
None of these
The parametric equations of the hyperbola \({x^{2}\over a^{2}}-{y^{2}\over b^{2}}=1\) are
- (a)
x = \(\sqrt2a\), y = b
- (b)
x = a tan \(\theta\), y = sec \(\theta\)
- (c)
x = a sec \(\theta\), y = b sin \(\theta\)
- (d)
None of these
If x=9 is the chord of contact of the hyperbola x2-y2=9 ,then equation of the corresponding pair of tangents is
- (a)
9x2-8y2+18x-9=0
- (b)
9x2-8y2-18x+9=0
- (c)
9x2-8y2-18x-9=0
- (d)
9x2-8y2+18x+9=0
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
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 \({x^{2}\over cos^{2}\alpha}-{-y^{2}\over sin^{2}\alpha}=1\), 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
If the angle between the lines represented by 6x2 + 5xy - 4y2 + 7x + 13y - 3 = 0 is tan-1(m) and a2 + b2 - ab - a - b + 1 \(\le \) 0, then 5a + 6b is equal to
- (a)
\(\frac{1}{m}\)
- (b)
m
- (c)
\(\frac{1}{2m}\)
- (d)
2m
The gradient of one of the lines ax2 + 2hxy + by2 = 0 is twice that of the other, then
- (a)
h2 = ab
- (b)
h = a + b
- (c)
8h2 = 9ab
- (d)
ah2 = 4ab
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
The pair of straight lines joining the origin to the common points of x2 + y2 = 4 and y = 3x + c are perpendicular, if c2 is equal to
- (a)
20
- (b)
13
- (c)
1/5
- (d)
5
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
If \({ x }^{ 2 }+\alpha { y }^{ 2 }+2\beta y={ a }^{ 2 }\) represents a pair of perpendicular straight lines, then
- (a)
\(\alpha =1,\beta =a\)
- (b)
\(\alpha =1,\beta =-a\)
- (c)
\(\alpha =-1,\beta =-a\)
- (d)
\(\alpha =-1,\beta =a\)