Mathematics - Conic Sections
Exam Duration: 45 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)
\({ x }^{ 2 }=-\frac { 9 }{ 2 } y\)
If the line 3y = 3x + 1 is a normal to the ellipse \(\frac { { x }^{ 2 } }{ 5 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\), then the length of the minor axis of the ellipse is
- (a)
\(4\quad or\quad \frac { 2 }{ 3 } \sqrt { 55 } \)
- (b)
\(2\quad or\quad \frac { 2 }{ 5 } \sqrt { 55 } \)
- (c)
\(3\quad or\quad \sqrt { 5 } \)
- (d)
\(11\quad or\quad \sqrt { 13 } \)
The equation of a tangent to the parabola y2 = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is
- (a)
(-1, 1)
- (b)
(0, 2)
- (c)
(2, 4)
- (d)
(-2, 0)
If f(x) = ax3 of bx2 + Cx + d, (a, b, c, d are rational nos.) and roots of f(x) = 0 are eccentricities of a parabola and a rectangular hyperbola then a + b + c + d equals
- (a)
-1
- (b)
0
- (c)
1
- (d)
data inadequate
The area of a triangle inscribed in an ellipse bears a constant ratio to the area of the triangle formed by joining points on the auxiliary circle corresponding to the vertices of the first triangle. This ratio is
- (a)
b/a
- (b)
2a/b
- (c)
a2/b2
- (d)
b2/a2
The diameter of 16x2 - 9y2 = l44, which is conjugate to x = 2y, is
- (a)
\(y=\frac { 16 }{ 9 } x\)
- (b)
\(y=\frac { 32 }{ 9 } x\)
- (c)
\(x=\frac { 16 }{ 9 } y\)
- (d)
\(x=\frac { 32 }{ 9 } y\)
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 radius of the circle passing through the foci of the \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } =1\) and having its center (0, 3) is
- (a)
4
- (b)
3
- (c)
\(\sqrt { 12 } \quad \)
- (d)
7/2
The equation 16x2 - 3y2 - 32x -12y - 44= 0 represents a hyperbola with
- (a)
length of the transverse axis = 2\(\sqrt3\)
- (b)
length of the conjugate axis = 8
- (c)
cente at (1, -2)
- (d)
eccentricity = \(\sqrt19\)
If (5, 12) and (24, 7) are the focii of a conic passing through the origin, then the eccentricity of conic is
- (a)
\(\frac { \sqrt { 386 } }{ 38 } \)
- (b)
\(\frac { \sqrt { 386 } }{ 12 } \)
- (c)
\(\frac { \sqrt { 386 } }{ 13 } \)
- (d)
\(\frac { \sqrt { 386 } }{ 25 } \)
Let us define a region R in xy-plane as a set of points (x, y) satisfying [x2] = [y] (where [x] denotes greatest integer \(\le \)x), then the region R defines
- (a)
a parabola whose axis is horizontal
- (b)
a parabola whose axis is vertical
- (c)
integer point on the parabola y = x 2
- (d)
none of the above
The condition that the straight line lx + my + n = 0 touches the parabola x2 = 4ay is
- (a)
bn = am2
- (b)
al2 - mn = 0
- (c)
In = am2
- (d)
am = In2
A line L passing through the focus of the parabola y2 = 4 (x - 1) intersects the parabola in two distinct points. If' m' be the slope of the line L, then
- (a)
\(m\in \left( -1,1 \right)\)
- (b)
\(m\in \left( -\infty ,-1 \right) \cup \left( 1,\infty \right)\)
- (c)
\(m\in R\)
- (d)
none of these
If the vertex of the parabola y=x2-16x+K lies on x-axis then the value of K is
- (a)
16
- (b)
8
- (c)
64
- (d)
-64
If the latus rectum of an ellipse with axis along x-axis and centre at origin is 10 distance between foci= length of minor axis, then the equation of the ellipse is
- (a)
\({x^2\over 25}+{y^2\over10}=1\)
- (b)
\({x^2\over 50}+{y^2\over100}=1\)
- (c)
\({x^2\over 5}+{y^2\over 5}=1\)
- (d)
\({x^2\over 100}+{y^2\over 50}=1\)
The equation \({x^2\over 14-a}+{y^2\over9-a}=1\) represents a/an
The equation \({x^2\over 1-r}-{y^2\over 1+r}=1, |r|<1\) represents a/an
- (a)
ellipse
- (b)
hyperbola
- (c)
circle
- (d)
None of these
The foci of a hyperbola coincide with the foci of the ellipse \({x^2\over 25}+{y^2\over9}=1\) Find the equation of the hyperbola if its eccentricity is 2.
- (a)
\({x^2\over 12}-{y^2\over4}=1\)
- (b)
\({x^2\over 4}-{y^2\over12}=1\)
- (c)
\({x^2\over 3}-{y^2\over4}=1\)
- (d)
\({x^2\over 4}-{y^2\over3}=1\)
Find the equation of the hyperbola whose foci are (0, ±12) and the length of the latus rectum is 36.
- (a)
3y2- x2 = 108
- (b)
3y2+ x2 = 108
- (c)
x2- 3y2 = 108
- (d)
3x2- y2 = 108
Statement-I: In an ellipse, if length of major axis is 26 and foci are (± 5, 0), then the equation of ellipse is \(\frac{x^2}{144}+\frac{y^2}{169}=1\)
Statement-II: In an ellipse, if length of minor axis is 16 and foci are (0, ± 6), then the equation of ellipse is \(\frac{x^2}{64}+\frac{y^2}{100}=1\)
- (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.