JEE Main Mathematics - Conic Sections
Exam Duration: 60 Mins Total Questions : 30
The equation of parabola having vertex (0, 0) passing through (2, 3) and axis is X - axis, is
- (a)
\({ x }^{ 2 }=\frac { 9 }{ 2 } y\)
- (b)
\({ y }^{ 2 }=\frac { 9 }{ 2 } x\)
- (c)
\({ y }^{ 2 }=-\frac { 9 }{ 2 } x\)
- (d)
\({ x }^{ 2 }=-\frac { 9 }{ 2 } y\)
From the point A(4, 3), tangents are drawn to the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } =1\) to touch the ellipse at B and C. EF is a tangent to the ellipse parallel to the line BC and toward the point A. The distance of A from EF is equal to.
- (a)
\(\frac { 4\sqrt { 11 } -5 }{ 2 } \)
- (b)
\(\frac { 24-4\sqrt { 18 } }{ 5 } \)
- (c)
\(\frac { 24+4\sqrt { 18 } }{ 5 } \)
- (d)
\(\frac { 24+2\sqrt { 11 } }{ 5 } \)
If in an ellipse, the distances between its foci is 6 and minor axis is 8. Then, its eccentricity is
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
\(\frac { 4 }{ 5 } \)
- (c)
\(\frac { 1 }{ \sqrt { 5 } } \)
- (d)
\(\frac { 3 }{ 5 } \)
The difference between the second-degree curve and pair of asymptotes is constant. If second-degree curve represented by a hyperbola S = 0, then the equation of its asymptotes is S + \(\lambda\) A = 0 where \(\lambda\) is constant. which will be a pair of straight lines, then we get \(\lambda\) . Then equation of asymptotes is A \(\equiv \) S + \(\lambda\) = 0 and if equation of conjugate hyperbola of S represented by S1 then A is the arithmetic mean of S and S1 If angle between the asymptotes of hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) \(\pi \over3\) then the eccentricity of conjugate hyperbola is
- (a)
\(\sqrt2\)
- (b)
2
- (c)
2/ \(\sqrt3\)
- (d)
4 / \(\sqrt3\)
The set of values of a for which (13x-1)2 + (13y - 2)2 = a (5x+12 y -1)2 represents an ellipse , if
- (a)
1 < a < 2
- (b)
0 < a < 1
- (c)
2 < a < 3
- (d)
None of these
If x = 9 is the chord of contact of the hyperbola X2 - Y2 = 9, then the equation of the corresponding pair of tangents is
- (a)
9x2 - 8y2 + 18x - 9 = O
- (b)
9x2 - 8y2 - 18x + 9 = 0
- (c)
9x2 - 8y2 - l8x - 9 = 0
- (d)
9x2 - 8y2 + 18x + 9 = O
The equation of a hyperbola, conjugate to the hyperbola x2 + 3xy + 2y2 + 2x + 3y = a is
- (a)
x2 + 3xy + 2y2 + 2x + 3y + 1 = 0
- (b)
X2 + 3xy + 2y2 + 2x + 3y + 2 = 0
- (c)
x2 + 3xy + 2y2 + 2x + 3y + 3 = 0
- (d)
x2 + 3xy + 2y2 + 2x + 3y + 4 = 0
If the tangent and normal to a rectangular hyperbola cut off intercepts X1 and X2 on one axis and y1 and y2 on the other axis, then
- (a)
X1Y1 + x2Y2 = 0
- (b)
x1Y2 + x2Y1 = 0
- (c)
x1x2 + y1y2 = 0
- (d)
none of these
For the ellipse \(\frac { x^{ 2 } }{ a^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\\ \\ \) the Equation of the diameter conjugate to ax -by= 0 is
- (a)
bx + ay = 0
- (b)
bx - ay = 0
- (c)
a3y + b3x = 0
- (d)
a3y - b3x = 0
If the normals at (xi, yi), i = 1, 2, 3, 4 on the rectangular hyperbola .xy = c2, meet atthe point (\(\alpha, \beta\)). The value of \(\sum { { y }_{ i } } \)
- (a)
c\(\beta\)
- (b)
c\(\alpha\)
- (c)
\(\alpha\)
- (d)
\(\beta\)
The eccentric angle of a point on the ellipse X2 + 3y2 = 6 at point on the ellipse X2 + 3Y2 = 6 at a distance 2 unit from origin is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { 3\pi }{ 4 } \)
- (c)
\(\frac { 5\pi }{ 4 } \)
- (d)
\(\frac { 7\pi }{ 4 } \)
The standard equation of an ellipse and the general equation of a circle are respectively given by the equations \(E:\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } -1=0\) and \(c:{ x }^{ 2 }+{ y }^{ 2 }2gx+2fy+c=0\)
then the Equation \(E\quad +\lambda C=o,\lambda \neq 0\) \(\left( \frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } -1 \right) +\lambda ({ x }^{ 2 }+{ y }^{ 2 }+2gx+2fy+c)=0\)represents a curve which passes through the common points of the ellipse and the circle Let the eccentric angles of three points p, Q and R in the Ellipse E are\(\frac { \pi }{ 2 } +a\) and \(\pi +a\) circle C through P, Q and R cuts the ellipse E again at S, then the eccentric angle of S is
- (a)
\(\frac { \pi }{ 2 } -3a\quad \)
- (b)
\(\pi -3a\)
- (c)
\(\frac { 3\pi }{ 2 } -3a\)
- (d)
\(2\pi -3a\)
If P and Q are the points (at12 ,2at1) and (at22 , 2at2) and normals at P and Q meet on the parabola y2 = 4ax, then t1t2 equals
- (a)
2
- (b)
-1
- (c)
-2
- (d)
-4
If (a, b) is the mid-point of chord passing through the vertex of the parabola y2 = 4x, then
- (a)
a=2b
- (b)
2a=b
- (c)
a2 = 2b
- (d)
2a = b2
The points on the axis of the parabola 3y2 + 4y - 6x + 8 = 0 from when 3 distinct normals can be drawn is given by
- (a)
\(\left( a,\frac { 4 }{ 3 } \right) ;a>\frac { 19 }{ 9 }\)
- (b)
\(\left( a,-\frac { 2 }{ 3 } \right) ;a>\frac { 19 }{ 9 }\)
- (c)
\(\left( a,\frac { 1 }{ 3 } \right) ,a>\frac { 7 }{ 9 }\)
- (d)
none of these
If a circle and a parabola intersect in 4 points, then the algebraic sum of the ordinates is
- (a)
proportional to the arithmetic mean of the radius and latus-rectum
- (b)
zero
- (c)
equal to the ratio of arithmetic mean and latus-rectum
- (d)
none of the above
the distance of the mid point of the line joining two points (4,0) and (0, 4) from the centre of the circle x2+y2=16 is
- (a)
√2
- (b)
2√2
- (c)
3√2
- (d)
2√3
find the coordinates of the foci and eccentricity respectively of the ellipse \({x^2\over 25}+{y^2\over 9}=1\)
- (a)
\((0, ±4),{4\over 5}\)
- (b)
\((±4,0),{4\over 5}\)
- (c)
\((0, ±4),{4\over3}\)
- (d)
\((0, ±2),{4\over5}\)
The length of latus rectum and vertices respectively of the ellipse 9x2+25y2=225
- (a)
\({18\over 5},(\pm5, 0)\)
- (b)
\({5\over 18},(\pm5, 0)\)
- (c)
\({5\over 6},(\pm5, 2)\)
- (d)
\({5\over 18},(0,\pm5 )\)
The point (at2, 2bt) lies on the hyperbola \({x^2\over a^2}-{y^2\over b^2}=1\) for
- (a)
all values of t
- (b)
t2=2+√5
- (c)
t2=2-√5
- (d)
no real values of t
If the eccentricity of the hyperbola x2 - y2 sec2 θ = 4 is √3 times the eccentricity of the ellipse x2 sec2 θ - Y2 = 16 then a value of θ is
- (a)
π/6
- (b)
3π/4
- (c)
π/3
- (d)
π/2
If the parabola y2 = 4ax passes through the point (3,2), then the length of its latus rectum is
- (a)
\(2\over3\)
- (b)
\(4\over3\)
- (c)
\(1\over3\)
- (d)
4
If the distances of foci and vel tex of hyperbola from the centre are c and a respectively, then
Statement-I: Eccentricity is always less than l.
Statement-II: Foci are at a distance of ae from the centre.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement-I
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement-I.
- (c)
If Statement-I is true but Statement-II is false
- (d)
If Statement-I is false and Statement-II is true.
Statement-I: In an ellipse, if ends of major axis are (± 3,0) and ends of minor axis are (0, ± 2), then the equation of ellipse is \(\frac{x^2}{9}+\frac{y^2}{4}=1\)
Statement-II: Points on major axis are (±a, 0) and points on minor axis are (0, ±b) if \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)\) is an equation of ellipse.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement-I
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement-I
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.