The locus of the passes through (a, 0), (- a, 0) is the centre of a circle which two variable points

• x = 1

• x + y = a

• x + y = 2a

• x = 0

The intercept on the line y = x by the circle x² = y² - 2x = 0 is AB. Equation of the circle with AB as the diameter is

• x² + y² = 1

• x(x - 1) + y(y - 1) = 0

• x² + y² = 2

• (x - 1)(x - 2) + (y - 1)(y - 2) = 0

If the coordinates of one end of a diameter of the circle x2 + y² + 4x-8y + 5 = 0 are (2, 1), the coordinates of the other end are

• (- 6, - 7)

• (6 , 7)

• (- 6, 7)

• (7, - 6)

The locus of the middle points of all chords of the parabola y² = 4ax passing through the vertex is

• a straight line

• an ellipse

• a parabola

• a circle

Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is $\left(- \frac{1}{2}, \frac{1}{2}\right)$

116.

If t₁ and t₂ be the parameters of the end points of a focal chord for the parabola y² = 4ax, then which one is true?

• t₁t₂ = 1

• $\frac{{\mathrm{t}}_1}{{\mathrm{t}}_2} = 1$

• t₁t₂ = - 1

• t₁ + t₂ = - 1

S and T are the foci of an ellipse and B is end point of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

For different values of a, the locus of the point of intersection of the two straight lines $\sqrt{3}\mathrm{x} - \mathrm{y} - 4\sqrt{3}\mathrm{\alpha } = 0$ and $\sqrt{3}\mathrm{\alpha x} + \mathrm{\alpha y} - 4\sqrt{3} = 0$

• a hyperbola with eccentricity 2

• an ellipse with eccentricity $\sqrt{\frac{2}{3}}$

• a hyperbola with eccentricity $\sqrt{\frac{19}{16}}$

• an ellipse with eccentricity $\frac{3}{4}$

The circles x² + y² - 10x + 16 = 0 and x² + y² = a² intersect at two distinct points, if

• a < 2

• 2 < a < 8

• a > 8

• a = 2

For the two circles x² + y² = 16 and x² + y² - 2y = 0 there is/are

• one pair of common tangents

• only one common tangent

• three common tangents

• no common tangent