Conic Section

A circle passes through the point (3, 4) and cuts the circle x² + y² = a² orthogonally; the locus of its centre is a straight line. If the distance of this straight line from the origin is 25, then a is equal to

• 250

• 225

• 100

• 25

The equation to the line joining the centres of the circles belonging to the coaxial system of circles 4x² + 4y² - 12x + 6y - 3 + $\mathrm{\lambda }$(x + 2y - 6) = 0 is

• 8x - 4y - 15 = 0

• 8x - 4y + 15 = 0

• 3x - 4y - 5 = 0

• 3x - 4y + 5 = 0

Let x + y = k be a normal to the parabola y² = 12x. If p is length of the perpendicular from the focus of the parabola onto this normal, then 4k - 2p² is equal to

• 1

• 0

• - 1

• 2

If the line 2x + 5y = 12 intersects the ellipse 4x² + 5y² = 20 in two distinct points A and B,then mid-point of AB is

• (0, 1)

• (1, 2)

• (1, 0)

• (2, 1)

Equation of one of the tangents passing through(2, 8) to the hyperbola 5x² - y² = 5 is

• 3x + y - 14 = 0

• 3x - y + 2 = 0

• x + y + 3 = 0

• x - y + 6 = 0

The area (in sq units) of the equilateral triangle formed by the tangent at ($\sqrt{3}$, 0) to the hyperbola x² - 3y² = 3 with the pair of asymptotes of the hyperbola is

• $\sqrt{2}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $2\sqrt{3}$

If the length of the tangent from (h, k) to the circle x² + y² = 16 is twice the length of the tangent from the same point to the circle x² + y² + 2x + 2y = 0, then

• ${\mathrm{h}}^2 + {\mathrm{k}}^2 + 4\mathrm{h} + 4\mathrm{k} + 16 = 0$

• ${\mathrm{h}}^2 + {\mathrm{k}}^2 + 3\mathrm{h} + 3\mathrm{k} = 0$

• $3{\mathrm{h}}^2 + 3{\mathrm{k}}^2 + 8\mathrm{h} + 8\mathrm{k} + 16 = 0$

• $3{\mathrm{h}}^2 + 3{\mathrm{k}}^2 + 4\mathrm{h} + 4\mathrm{k} + 16 = 0$

($\mathrm{\alpha }$, 0) and (b, 0) are centres of two circles belonging to a coaxial system of which y-axis is the radical axis. If radius of one of the circles is 'r', then the radius of the other circle is

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 + {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 - {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 - {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• ${\left({\mathrm{r}}^2 + {\mathrm{b}}^2 + {\mathrm{a}}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

If the circle x² + y² + 4x - 6y + c =0 bisects the circumference of the circle x² + y² - 6x + 4y - 12 = 0, then c is equal to

• 16

• 24

• - 42

• - 62

A circle of radius 4, drawn on a chord of the parabola y² = 8x as diameter, touches the axis of the parabola. Then, the slope of the chord is

• $\frac{1}{2}$

• $\frac{3}{4}$

• 1

• 2