Mathematics : Conic Section

The slopes of the focal chords of the parabola y² = 32x, which are tangents to the circle x² + y² = 4, are

• $\frac{1}{2}, - \frac{1}{2}$

• $\frac{1}{\sqrt{3}}, \frac{- 1}{\sqrt{3}}$

• $\frac{1}{\sqrt{15}}, \frac{- 1}{\sqrt{15}}$

• $\frac{2}{\sqrt{5}}, \frac{- 2}{\sqrt{5}}$

If tangents are drawn from any point on the circle x² + y² = 25 to the ellipse $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$,  then the angle between the tangents is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

An ellipse passing through has $\left(4\sqrt{2}, 2\sqrt{6}\right)$ foci at (- 4, 0) and (4, 0). Then, its eccentricity

• $\sqrt{2}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{3}}$

A hyperbola passing through a focus of the  ellipse$\frac{{\mathrm{x}}^2}{169} + \frac{{\mathrm{y}}^2}{25} = 1$. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of eccentricities is 1. Then, the equation of the hyperbola is

• $\frac{{\mathrm{x}}^2}{144} - \frac{{\mathrm{y}}^2}{9} = 1$

• $\frac{{\mathrm{x}}^2}{169} - \frac{{\mathrm{y}}^2}{25} = 1$

• $\frac{{\mathrm{x}}^2}{144} - \frac{{\mathrm{y}}^2}{25} = 1$

• $\frac{{\mathrm{x}}^2}{125} - \frac{{\mathrm{y}}^2}{9} = 1$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{curves} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{and} \frac{{\mathrm{x}}^2}{25} + \frac{{\mathrm{y}}^2}{16} = 1 \\ \mathrm{cut} \mathrm{each} \mathrm{other} \mathrm{orthogonally}, \mathrm{then} {\mathrm{a}}^2 - {\mathrm{b}}^2 = ? \end{gathered}$$

• 9

• 400

• 75

• 41

Te area (m sq units) of the region bounded by x = -1, x = 2, y = x² + 1 and y = 2x - 2 is

• 10

• 7

• 8

• 9

The sum of the minimum and maximum distance of the point (4, - 3) to the circle x² + y² + 4x - 10y - 7 = 0, is

• 10

• 12

• 16

• 20

The locus of centres of the circles, which cut the circles x² + y² + 4x - 6y + 9 and x² + y² - 5x + 4y + 2 = 0 orthogonally, is

• 3x + 4y - 5 = 0

• 9x - 10y + 7 = 0

• 9x + 10y - 7 = 0

• 9x - 10y + 11 = 0

If x - y + 1 = 0 meets the circle x² + y² + y - 1 = 0 at A and B, then the equation of the circle with AB as diameter is

• 2(x² + y²) + 3x - y + 1 = 0

• 2(x² + y²) + 3x - y + 2 = 0

• 2(x² + y²) + 3x - y + 3 = 0

• x² + y² + 3x - y + 4 = 0

An equilateral triangle is inscribed in the parabola y² = Bx, with one of its vertices is the vertex of the parabola. Then, length of the side of that triangle is

• $24\sqrt{3} \mathrm{units}$

• $16\sqrt{3}$ units

• $8\sqrt{3} \mathrm{units}$

• $4\sqrt{3} \mathrm{units}$