Mathematics - Conic Sections
Exam Duration: 45 Mins Total Questions : 30
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 normal at P to a hyperbola of eccentricity e, intersects its transverse and conjugate axes at L and M respectively. If locus of the mid point of LM is hyperbola, then
eccen.triCi~ of the hyperbola is
- (a)
\(\left( \frac { e+1 }{ e-1 } \right) \)
- (b)
\(\frac { e }{ \sqrt { ({ e }^{ 2 }-1) } } \)
- (c)
e
- (d)
none of these
If a source of light is placed at the fixed point of a parabola and if the parabola is a reflecting surface, then the ray will bounce back in a line parallel to the axis of the parabola. A ray of light is coming along the line y=2 from the positive direction of x-axis and strikes a concave mirror whose intersection with the x - y plane is a parabola y2=8x, then the equation of the reflected ray is
- (a)
2x+5y=4
- (b)
3x+2y=6
- (c)
4x+3y=8
- (d)
5x+4y=10
The line x cas a .+ y sin a = p touches the hyperbola
- (a)
a2 cos2 \(\alpha\) _.b2 sin2 \(\alpha\)= p2
- (b)
a2 cos2 \(\alpha\) - b2 sin2 \(\alpha\)= p
- (c)
a2 cos2 \(\alpha\) + b2 sin2 \(\alpha\)= p2
- (d)
a2 cos2 \(\alpha\) + b2 sin2 \(\alpha\)= p
The connectivity of an ellipse \(\frac { { x }^{ 2 } }{ { { a }^{ 2 } } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) whose latusrectum is half of its minor axis is
- (a)
\(\frac { 1 }{ \sqrt { 2 } } \)
- (b)
\(\sqrt { \frac { 2 }{ 3 } } \)
- (c)
\(\frac { \sqrt { 3 } }{ 2 } \)
- (d)
none of these
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
The equation of tangent parallel to y = x drawn to \(\frac { { x }^{ 2 } }{ 3 } -\frac { { y }^{ 2 } }{ 2 } =1\) is
- (a)
x - y + 1 = 0
- (b)
x - y - 2 = 0
- (c)
x + y - 1 = 0
- (d)
x - y - 1 = 0
If the tangent to the ellipse X2 + 4y2 = 16 at the point p(\(\theta\)) is a normal to the circle x2 + y2 - 8x - 4y = 0, then \(\theta\)
- (a)
\(\pi/2\)
- (b)
\(\pi/4\)
- (c)
0
- (d)
\(-\pi/4\)
If the 4th term in the expansion of \({ \left( px+\frac { 1 }{ x } \right) }^{ n },n\epsilon N\quad is\quad \frac { 5 }{ 2 } \) and three normals to the parabola y2 = x are drawn through a point (q, 0), then
- (a)
q = p
- (b)
q > p
- (c)
q < p
- (d)
pq = 1
Two perpendicular tangents PA and PB are drawn to y2 = 4ax, minimum length of AB is equal to
- (a)
a
- (b)
4a
- (c)
8a
- (d)
2a
A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is (y - 2)2 = 4 (x + 1). After reflection, the ray must pass through the point
- (a)
(-2,0)
- (b)
(-1,2)
- (c)
(0,2)
- (d)
(2,0)
The equation \(\sqrt { \left\{ { \left( x-3 \right) }^{ 2 }+{ \left( y-1 \right) }^{ 2 } \right\} } +\sqrt { \left\{ { \left( x+3 \right) }^{ 2 }+{ \left( y-1 \right) }^{ 2 } \right\} } =6\) represents
- (a)
an ellipse
- (b)
a pair of straight lines
- (c)
a circle
- (d)
a straight line joining the point (-3,1) to the point (3,1)
If the tangent at P on y2 = 4ax meets the tangent at the vertex in Q and S is the focus of the parabola, then \(\angle\)SQP is equal to
- (a)
\({ \pi }/{ 3 }\)
- (b)
\({ \pi }/4\)
- (c)
\({ \pi }/2\)
- (d)
\({ 2\pi }/{ 3 }\)
Find the centre and be the radius of the circle x2+y28x+10y-8=0
- (a)
(5,4), -7
- (b)
(-4, -5),7
- (c)
(4,-5),7
- (d)
(5,-4), 7
If the line x-1=0 is directrix of the parabola y2-kx+8=0, then one of the values k is
- (a)
1/8
- (b)
8
- (c)
4
- (d)
1/4
the equation of the directrix of the parabola x2-4x-3y+10=0 is
- (a)
y=-5/4
- (b)
y=5/4
- (c)
y=-3/4
- (d)
x=5/4
If for the ellipse \({x^2\over a^2}+{y^2\over b^2}=1\) y-axis is the minor axis and the length of the latus rectum is one half of the length of its minor axis, then its eccentricity is
- (a)
\(1\over \sqrt 2\)
- (b)
1/2
- (c)
\(\sqrt3\over 2\)
- (d)
3/4
The centre of the ellipse 9x2+25y2-18x-100y-116=0 is
- (a)
(1,1)
- (b)
(-1,2)
- (c)
(1,2)
- (d)
(2,2)
The eccentricity of the hyperbola \({x^2\over a^2}-{y^2\over b^2}=1\) which passes through the points (3,0) and (3,√2) is
- (a)
\(1\over \sqrt{13}\)
- (b)
\(\sqrt{13}\)
- (c)
\( \sqrt{13}\over 2\)
- (d)
\( \sqrt{13}\over 3\)
Find the focii of the hyperbola with vertices at (0,±6) and \(e={5\over 3}\)
- (a)
(±10,0)
- (b)
(10, 0)
- (c)
(0, ±10)
- (d)
(10,-10)
Statement-I: The sum of focal distances of a point on the ellipse 9x2 + 4y2 - 18x - 24y + 9 = 0 is 4.
Statement-II:The equation 9x2 +4y2 -18x-24y+9=0 can be expressed as 9(x - 1)2 + 4(y - 3)2 = 36.
- (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.
The sum of the minimum distance and the maximum distance from the point (4, - 3) to the circle x2 + y2 + 4x - 10y - 7 = 0 is
- (a)
20
- (b)
12
- (c)
10
- (d)
16