The eccentricity of an ellipse, with its centre at the origin, is 1 /2 . If one of the directrices is x = 4, then the equation of the ellipse is

• 3x² +4y² = 1

• 3x²+ 4y² = 12

• 4x² +3y² = 12

• 4x² +3y² = 12

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45^o, 30^o and 30^o, then the height of the tower (in m) is

• 50√2

• 100

• 50

• 100√3

Tangents are drawn to the hyperbola 4x² – y² = 36 at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of △PTQ is

• 36√5

• 45√5

• 54√3

• 60√3

44.

Tangent and normal are drawn at P(16, 16) on the parabola y² = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and,∠CPB = θ then a value of tan θ is

• 4/3

• 1/2

• 2

• 3

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is

• 3x + 2y = 6xy

• 3x + 2y = 6

• 2x + 3y = xy

• 3x + 2y = xy

Let P be the foot of the perpendicular from focus S of hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ on the line bx- ay = 0 and let C be the centre of the hyperbola. Then, the area of the rectangle whose sides are equal to that of SP and CP is

• 2ab

• ab

• $\frac{\left({\mathrm{a}}^{2 }+ {\mathrm{b}}^2\right)}{2}$

• $\frac{\mathrm{a}}{\mathrm{b}}$

B is an extremity of the minor axis of an ellipse whose foci are S and S'. If $\angle$SBS' is a right angle, then the eccentricity of the ellipse is

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{2}{3}$

• $\frac{1}{3}$

The axis of the parabola x² + 2xy + y² - 5x + 5y - 5 = 0 is

• x + y = 0

• x + y - 1 = 0

• x - y + 1 = 0

• $\mathrm{x} - \mathrm{y} = \frac{1}{\sqrt{2}}$

The line segment joining the foci of the hyperbola x^{2 -} y² + 1 = 0 is one of the diameters of a circle. The equation of the circle is

• x² + y² = 4

• x² + y² = $\sqrt{2}$

• x² + y² = 2

• x² + y² = $2\sqrt{2}$

If one of the diameters of the curve x² + y² - 4x - 6y + 9 = 0 is a chord of a circle with centre (1, 1), the radius of this circle is

• 3

• 2

• $\sqrt{2}$

• 1