Prove that for all values of m, except zero the straight line

y =  mx + $\frac{\mathrm{a}}{\mathrm{m}}$ touches the parabola y² = 4ax.

142.

If the lines 3x - 4y - 7 = 0 and 2x - 3y - 5 = 0 are two diameters of a circle of area 49$\mathrm{\pi }$ sq unit, then equation of the circle is

• x² + y² + 2x - 2y - 62 = 0

• x² + y² - 2x + 2y - 62 = 0

• x² + y² - 2x + 2y - 47 = 0

• x² + y² + 2x - 2y - 47 = 0

The locus of middle point of chords of hyperbola 3x² - 2y² + 4x - 6y = 0 parallel to y = 2x is

• 3x - 4y = 4

• 3x - 4x + 4 = 0

• 4x - 3y = 3

• 3x - 4y = 2

The equation of the common tangent touching the circle (x - 3)² + y² = 9 and parabola y = 4x above the x-axis is

• √3y = 3x + 1

• √3y = -( x + 3 )

• √3y = x + 3

• √3y = -( 3x + 1 )

The equation of the tangents to the ellipse 4x² + 3y² = 5 which are parallel to the line y = 3x + 7 are

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{\frac{155}{3}}$

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{\frac{155}{12}}$

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{\frac{95}{12}}$

• None of these

The radius of the circle passing through the foci of the ellipse $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$ and having its centre (0, 3) is

• 4

• 3

• $\sqrt{12}$

• 7/2

The equation of the circle on the common chord of the circles (x - a)² + y² = a² and x² + (y + b)² = b² as diameter is

• x² + y² = 2ab(bx + ay)

• x² + y² = bx + ay

• (a² + b²)(x² + y²) = 2ab(bx - ay)

• (a² + b²)(x² + y²) = 2(bx + ay)

The eccentricity of the hyperbola which passes through (3, 0) and (3$\sqrt{2}$, 2) is

• $\sqrt{\left(13\right)}$

• $\frac{\sqrt{13}}{3}$

• $\sqrt{\frac{13}{4}}$

• None of these

The line x - 1 = 0 is the directrix of the parabola y² - kx + 8 = 0. Then, one of the value of k is

• $\frac{1}{8}$

• 8

• 4

• $\frac{1}{4}$

The curve represented by

$\mathrm{x} = 3\left(\cos \left(\mathrm{t}\right) + \sin \left(\mathrm{t}\right)\right), \mathrm{y} = 4\left(\cos \left(\mathrm{t}\right) - \sin \left(\mathrm{t}\right)\right)$ is

• ellipse

• parabola

• hyperbola

• circle