1
The equation of the circle passing through (1,5) and (4,1) and touching \(y-\)axis is
\[ x^2 + y^2 - 5x - 6y + 9 + \lambda (4x + 3y - 19) = 0 \]
where \(\lambda\) is equal to
1
\(0, -\frac{40}{9}\)
2
\(0\)
3
\(\frac{40}{9}\)
4
\(-\frac{40}{9}\)
2
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
1
\(\frac{4}{3}\)
2
\(\frac{4}{\sqrt{3}}\)
3
\(\frac{2}{\sqrt{3}}\)
4
\(\frac{3}{2}\)
3
The circle \(x^2 + y^2 = 4x + 8y + 5\) intersects the line \(3x - 4y = m\) at two distinct points if
1
\(15 < m < 65\)
2
\(35 < m < 85\)
3
\(-85 < m < -35\)
4
\(-35 < m < 15\)
4
The length of the diameter of the circle which touches the \(x-\)axis at the point (1,0) and passes through the point (2,3) is
1
\(\frac{6}{5}\)
2
\(\frac{5}{3}\)
3
\(\frac{10}{3}\)
4
\(\frac{3}{5}\)
5
The radius of the circle \(3x^2 + by^2 + 4bx - 6by + b^2 = 0\) is
1
1
2
3
3
\(\sqrt{10}\)
4
\(\sqrt{11}\)
6
The centre of the circle inscribed in a square formed by the lines \(x^2 - 8x - 12 = 0\) and \(y^2 - 14y + 45 = 0\) is
1
(4,7)
2
(7,4)
3
(9,4)
4
(4,9)
7
The equation of the normal to the circle \(x^{2}+y^{2}-2x-2y+1=0\) which is parallel to the line \(2x+4y=3\) is
1
\(x+2y=3\)
2
\(x+2y+3=0\)
3
\(2x+4y+3=0\)
4
\(x-2y+3=0\)
8
If \(P(x,y)\) be any point on \(16x^{2}+25y^{2}=400\) with foci \(F_{1}(3,0)\) and \(F_{2}(-3,0)\) then \(PF_{1}+PF_{2}\) is
9
The radius of the circle passing through the point(6,2) two of whose diameter are \(x+y=6\) and \(x+2y=4\) is
1
10
2
\(2\sqrt{5}\)
3
6
4
4
10
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) is
1
\(4(a^{2}+b^{2})\)
2
\(2(a^{2}+b^{2})\)
3
\(a^{2}+b^{2}\)
4
\(\frac{1}{2}(a^{2}+b^{2})\)
11
If the normals of the parabola \(y^{2}=4x\) drawn at the end points of its latus rectum are tangents to the circle \((x-3)^{2}+(y+2)^{2}=r^{2}\), then the value of \(r^{2}\) is
12
If \(x+y=k\) is a normal to the parabola \(y^{2}=12x\), then the value of \(k\) is
13
The ellipse \(E_{1}:\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point(0,4) circumscribes the rectangle R. The eccentricity of the ellipse is
1
\(\frac{\sqrt{2}}{2}\)
2
\(\frac{\sqrt{3}}{2}\)
3
\(\frac{1}{2}\)
4
\(\frac{3}{4}\)
14
Tangents are drawn to the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) parallel to the straight line \(2x-y=1\). One of the points of contact of tangents on the hyperbola is
1
\(\left(\frac{9}{2\sqrt{2}},-\frac{-1}{\sqrt{2}}\right)\)
2
\(\left(\frac{-9}{2\sqrt{2}},\frac{1}{\sqrt{2}}\right)\)
3
\(\left(\frac{9}{2\sqrt{2}},\frac{1}{\sqrt{2}}\right)\)
4
\(\left(3\sqrt{3},-2\sqrt{2}\right)\)
15
The equation of the circle passing through the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) having centre at (0,3) is
1
\(x^{2}+y^{2}-6y-7=0\)
2
\(x^{2}+y^{2}-6y+7=0\)
3
\(x^{2}+y^{2}-6y-5=0\)
4
\(x^{2}+y^{2}-6y+5=0\)
16
Let \(C\) be the circle with centre at(1,1) and radius = 1. If \(T\) is the circle centered at(0,\(y\)) passing through the origin and touching the circle \(C\) externally, then the radius of \(T\) is equal to
1
\(\frac{\sqrt{3}}{\sqrt{2}}\)
2
\(\frac{\sqrt{3}}{2}\)
3
\(\frac{1}{2}\)
4
\(\frac{1}{4}\)
17
Consider an ellipse whose centre is of the origin and its major axis is along x-axis. If its eccentricity is \(\frac{3}{5}\) and the distance between its foci is 6, then the area of the quadrilateral inscribed in the ellipse with diagonals as major and minor axis of the ellipse is
18
Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is
1
\(2ab\)
2
\(ab\)
3
\(\sqrt{ab}\)
4
\(\frac{a}{b}\)
19
An ellipse has \(OB\) as semi minor axes, \(F\) and \(F^{\prime}\) its foci and the angle \(FBF^{\prime}\) is a right angle. Then the eccentricity of the ellipse is
1
\(\frac{1}{\sqrt{2}}\)
2
\(\frac{1}{2}\)
3
\(\frac{1}{4}\)
4
\(\frac{1}{\sqrt{3}}\)
20
The eccentricity of the ellipse \((x-3)^{2}+(y-4)^{2}=\frac{y^{2}}{9}\) is
1
\(\frac{\sqrt{3}}{2}\)
2
\(\frac{1}{3}\)
3
\(\frac{1}{3\sqrt{2}}\)
4
\(\frac{1}{\sqrt{3}}\)
21
If the two tangents drawn from a point \(P\) to the parabola \(y^{2}=4x\) are at right angles then the locus of \(P\) is
1
\(2x+1=0\)
2
\(x=-1\)
3
\(2x-1=0\)
4
\(x=1\)
22
The circle passing through(1,-2) and touching the axis of \(x\) at (3,0) passing through the point
1
\((-5,2)\)
2
\((2,-5)\)
3
\((5,-2)\)
4
\((-2,5)\)
23
The locus of a point whose distance from \((-2,0)\) is \(\frac{2}{3}\) times its distance from the line \(x=\frac{-9}{2}\) is
1
a parabola
2
a hyperbola
3
an ellipse
4
a circle
24
The values of \(m\) for which the line \(y=mx+2\sqrt{5}\) touches the hyperbola \(16x^{2}-9y^{2}=144\) are the roots of \(x^{2}-(a+b)x-4=0\), then the value of \((a+b)\) is
25
If the coordinates at one end of a diameter of the circle \(x^{2}+y^{2}-8x-4y+c=0\) are (1,2), the coordinates of the other end are
1
\((-5,2)\)
2
\((-3,2)\)
3
\((5,-2)\)
4
\((-2,5)\)