Mathematics - Coordinate Geometry II 1
Exam Duration: 45 Mins Total Questions : 30
If the line x-1=0,is the directrix of the parabola y2-kx+8=0,then one of the value of k is
- (a)
\(1\over8\)
- (b)
8
- (c)
\(1\over4\)
- (d)
4
The coordinates of the point of intersection of tangents at the points(x1,y1) and (x2,y2) on the parabola is (x3,y3);then y1,y2,y3 are in
- (a)
A.P
- (b)
G.P
- (c)
H.P
- (d)
None of these
If (x1,y1),(x2,y2) and (x3,y3) be three points on the parabola y2 =4ax,the normals at which meet in a points then value of y1+y2 +y3 is
- (a)
0
- (b)
1
- (c)
-1
- (d)
None of these
If x+y =k is a normal to the parabola y2 =12x,then k equal
- (a)
3
- (b)
9
- (c)
-9
- (d)
-3
Consider a circle with centre lying on the focus of the parabola y2 =2px such that it touches the directrix of the parabola .Then the point of intersection of the circle and the parabola,is
- (a)
(p,p)
- (b)
(p/2,-p)
- (c)
(-p/2,p)
- (d)
(-p/2,-p)
The eccentricity of the ellipse 16x2+7y2=112 is
- (a)
\(4\over3\)
- (b)
\(7\over16\)
- (c)
\(3\over7\)
- (d)
\(3\over4\)
The eccentricity of the ellipse represented by the equation 25x2+16y2-50x-175=0 is
- (a)
\(2\over5\)
- (b)
\(3\over5\)
- (c)
\(4\over5\)
- (d)
None of these
The straight line y=4x+c is tangent to the ellipse \({x^{2}\over8}+{y^{2}\over4}=1\) Then c is equal to
- (a)
\(\pm4\)
- (b)
\(\pm6\)
- (c)
\(\pm1\)
- (d)
\(\pm132\)
The coordinates of the foci of the ellipse represented by the equation in example (49) are
- (a)
(3,\(\pm3\))
- (b)
(2,\(\pm2\))
- (c)
(1,\(\pm1\))
- (d)
None of these
If S and S' are then foci of the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) and P(x,y) a point on it ,then value of SP+SP' is
- (a)
2a
- (b)
2b
- (c)
a+b
- (d)
a-b
The centre of the ellipse \({(x+y-2)^{2}\over 2}+{(x-y)^{2}\over16}=1\) is
- (a)
(0,0)
- (b)
(1,1)
- (c)
(0,1)
- (d)
(1,0)
The equation \({x^{2}\over 1-r}+{y^{2}\over 1+r}=1 \ (r>1)\) represents
- (a)
a circle
- (b)
a hyperbola
- (c)
an ellipse
- (d)
None of these
On the ellipse 4x2+9y2=1,the points at which the tangents are parallel to the line 8x=9y are
- (a)
\(({2\over5},{1\over5})\)
- (b)
\((-{3\over5},{2\over5})\)
- (c)
\((-{2\over5},{-1\over5})\)
- (d)
\(({2\over5},{-1\over5}) and (-{2\over5},{1\over5})\)
Let P be a variable point on the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) with foci F1 and F2 .If A is the triangle pF1F2.then the maximum value of A is
- (a)
\(ea\over b\)
- (b)
\(aeb\)
- (c)
\(ab\over e\)
- (d)
e/ab
The area of the quadrilateral formed by the tangents at the ends of latus rectum of the ellipse \({x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\) is
- (a)
\(4a^{2}\over e^{2}\)
- (b)
\(2a^{2}\over e^{2}\)
- (c)
\(2a^{2}\over e\)
- (d)
\(a^{2}\over e\)
The coordinates of the foci of the hyperbola 16x2-y2+64x+4y+44=0 is
- (a)
\((\pm \sqrt17-2.2)\)
- (b)
\((\pm \sqrt17+2.2)\)
- (c)
\((\pm \sqrt17-2)\)
- (d)
\((\pm \sqrt17-2)\)
The equation 9x2-16y2-18x-64y-199=0 represents
- (a)
a parabola
- (b)
an ellipse
- (c)
a hyperbola
- (d)
a pair of lines
The foci of the ellipse \({x^{2}\over 16^{2}}+{y^{2}\over b^{2}}=1\) and hyperbola \({x^{2}\over 144}+{y^{2}\over 81}={1\over25}\), coincide .Then value value b2 is
- (a)
1
- (b)
5
- (c)
7
- (d)
9
The equation of the hyperbola reffered to its axes as the axis of coordinates and whose distance between the two foci is 16 and ecentricity is \(\sqrt2\) is
- (a)
x2-y2=32
- (b)
y2-x2=32
- (c)
32x2-y2=1
- (d)
32y2-x2=1
The ecentricity of a rectangular hyperbola is always
- (a)
2
- (b)
\(\sqrt3\)
- (c)
\(\sqrt2\)
- (d)
None of these
Product of the perpendiculars from \(\left( \alpha ,\beta \right) \) to the lines ax2 + 2hxy + by2 = 0 is
- (a)
\(\frac { \left| a{ \alpha }^{ 2 }-2h\alpha \beta +b{ \beta }^{ 2 } \right| }{ \sqrt { \left\{ 4{ h }^{ 2 }+{ \left( a+b \right) }^{ 2 } \right\} } } \)
- (b)
\(\frac { \left| a{ \alpha }^{ 2 }-2h\alpha \beta +b{ \beta }^{ 2 } \right| }{ \sqrt { \left\{ 4{ h }^{ 2 }-{ \left( a-b \right) }^{ 2 } \right\} } } \)
- (c)
\(\frac { \left| a{ \alpha }^{ 2 }-2h\alpha \beta +b{ \beta }^{ 2 } \right| }{ \sqrt { \left\{ 4{ h }^{ 2 }-{ \left( a+b \right) }^{ 2 } \right\} } } \)
- (d)
none of these
The equation ax2 + by2 + cx + cy = 0 represents a pair of straight lines, if
- (a)
a + b = 0
- (b)
c = 0
- (c)
a + c = 0
- (d)
c(a + b) = 0
If the angle between the lines x2 - xy + ay2 = 0 is 45°. then value(s) of a is/are
- (a)
- 6
- (b)
0
- (c)
6
- (d)
12
If the lines represented by 2x2 - 5xy + 2y2 = 0 be the two sides of a parallelogram and the line 5x + 2y = 1 be one of its diagonal.
On the basis of above information, answer the following questions:
The centroid of the parallelogram is
- (a)
\(\left( \frac { 5 }{ 72 } ,\frac { 11 }{ 36 } \right) \)
- (b)
\(\left( \frac { 11 }{ 72 } ,\frac { 5 }{ 36 } \right) \)
- (c)
\(\left( \frac { 5 }{ 36 } ,\frac { 11 }{ 72 } \right) \)
- (d)
\(\left( \frac { 11 }{ 36 } ,\frac { 5 }{ 72 } \right) \)
If the lines represented by 2x2 - 5xy + 2y2 = 0 be the two sides of a parallelogram and the line 5x + 2y = 1 be one of its diagonal.
On the basis of above information, answer the following questions:
The area of the parallelogram is
- (a)
\(\frac{1}{36}\) sq unit
- (b)
\(\frac{1}{18}\) sq unit
- (c)
\(\frac{1}{9}\) sq unit
- (d)
none of these
If the lines represented by 2x2 - 5xy + 2y2 = 0 be the two sides of a parallelogram and the line 5x + 2y = 1 be one of its diagonal.
On the basis of above information, answer the following questions:
The ratio of the longer side to smaller side is
- (a)
6 : 5
- (b)
7 : 6
- (c)
5 :4
- (d)
4 : 3
If the lines represented by 2x2 - 5xy + 2y2 = 0 be the two sides of a parallelogram and the line 5x + 2y = 1 be one of its diagonal.
On the basis of above information, answer the following questions:
The ratio of the longer diagonal to smaller diagonal is
- (a)
6 : 5
- (b)
9 : 2
- (c)
13 : 5
- (d)
none of these
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:
Equation f2(x, y) = 0 is
- (a)
hx2 - (a - b)xy + hy2 = 0
- (b)
hx2 - (a - b)xy - hy2 = 0
- (c)
hx2 + (a - b)xy + hy2 = 0
- (d)
hx2 + (a - b)xy - hy2 = 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:
Equation f3(x, y) = 0 is
- (a)
(a - b)x2 - 4hxy + (a - b)y2 = 0
- (b)
( a - b )x2 - 4hxy - (a - b)y2 = 0
- (c)
( a - b )x2 + 4hxy - (b - a)y2 = 0
- (d)
( a - b )x2 + 4hxy - (a - b)y2 = 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 f2 (x, y) = 0 is
- (a)
bx2 - 2hxy + ay2 = 0
- (b)
ax2 + 2hxy + by2 = 0
- (c)
ax2 - 2hxy + by2 = 0
- (d)
none of the above