The minimum area of the triangle formed by any tangent to the ellipse ( x²/a² ) + ( y²/b² ) = 1 with the coordinate axes is

• a² + b²

• ( a + b )²/2

• ab

• ( a - b )²/2

If the line lx + my - n = 0 will be a normal to the hyperbola, then $\frac{{\mathrm{a}}^2}{{\mathrm{l}}^2} - \frac{{\mathrm{b}}^2}{{\mathrm{m}}^2} = \frac{{\displaystyle {\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2}}{\mathrm{k}}$, where k is equal to

• n

• n²

• n³

• None of these

Equation of the chord of the hyperbola 25x²  - 16y² = 400 which is bisected at the point (6, 2), is

• 6x - 7y = 418

• 75x - 16y = 418

• 25x - 4y = 400

• None of these

The centres of a set of circles, each of radius 3, lie on the circles x² + y² = 25. the locus of any point in the set is

• $4 \le {\mathrm{x}}^2 +\mathrm{\_}\hspace{0.17em}{\mathrm{y}}^2 \le 64$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \le 25$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \ge 25$

• $3 \le {\mathrm{x}}^2 + {\mathrm{y}}^2 \le 9$

The angle of intersection of the circles x² + y² - x + y - 8 = 0 and x² + y² + 2x + 2y - 11 = 0 is

• ${\tan }^{-1}\left(\frac{19}{9}\right)$

• ${\tan }^{-1}\left(19\right)$

• ${\tan }^{-1}\left(\frac{9}{19}\right)$

• ${\tan }^{-1}\left(9\right)$

If a tangent having slope of - $\frac{4}{3}$ to the ellipse $\frac{{\mathrm{x}}^2}{18} + \frac{{\mathrm{y}}^2}{32} = 1$ intersects the major and minor axes in points A and B respectively, then the area of $∆\mathrm{OAB}$ is equal to (O is centre of the ellipse)

• 12 sq units

• 48 sq units

• 64 sq units

• 24 sq units

167.

The locus of mid-points of tangents intercepted between the axes of ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ will be

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 1$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 2$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 3$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2} + \frac{{\mathrm{b}}^2}{{\mathrm{y}}^2} = 4$

If PQ is a double ordinate of hyperbola (x²/a²) - (y²/b²) = 1 such that OPQ is a equilateral triangle, O being the centre of the hyperbola, then the eccentricity 'e' of the hyperbola satisfies

• 1 < e < 2/√3

• e = 2/√3

• e = √3/2

• e > 2/√3

The lines 2x - 3y - 5 = 0 and 3x - 4y = 7 are diameters of a circle of area 154 sq units, then the equation of the circle is

• x² + y² + 2x - 2y - 62 = 0

• x² + y² + 2x - 2y - 47 = 0

• x² + y² - 2x + 2y - 47 = 0

• x² + y² - 2x + 2y - 62 = 0

The angle of depressions of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are θ and ∅ respectively, then the distance between their tops when tan θ = 4/3 and tan ∅ = 5/2 is

• 150/√3 m

• 100√3 m

• 150 m

• 100 m