If the algebraic sum of the perpendicular distances from the points (2, 0), (0, 2) and (1, 1) on a variable line is zero, then the line will pass through the fixed point

• (1, 2)

• (1, 1)

• (0, 0)

• (2, 1)

12.

The locus ofthe point of intersection of the lines $\mathrm{xcos}\left(\mathrm{\alpha }\right) + \mathrm{ysin}\left(\mathrm{\alpha }\right) = \mathrm{p}$ and $\mathrm{xsin}\left(\mathrm{\alpha }\right) - \mathrm{ycos}\left(\mathrm{\alpha }\right) = \mathrm{q}$ ($\mathrm{\alpha }$ is a variable) will be

• a circle

• a staright line

• a parabola

• an ellipse

The locus of the mid points of the chords of a circle which subtend a right angle at its centre (equation ofthe circle is x² + y² = a²)will be

• x² + y² = 3a²

• x² + y² = $\frac{{\mathrm{a}}^2}{3}$

• 2(x² + y²) = a²

• 4(x² + y²) = a²

If the line 3x - 2y + p = 0 is normal to the circle x² + y² = 2x - 4y - 1, then p will be

• - 5

• 7

• - 7

• 5

If the two circles x² + y² = r² and x² + y² - 10x + 16 = 0 intersect at two real points, then

• 1 < r < 7

• 3 < r < 10

• 2 < r < 9

• 2 < r < 8

The equation of the common tangent to the parabolas y² = 2x and x² = 16y will be

• x + y + 2 = 0

• x - 3y + 1 = 0

• x + 2y - 2 = 0

• x + 2y + 2 = 0

The equation of the tangent to the parabola y² = 8x, which is parallel to the line 2x - y + 7 = 0, will be

• y = x + 1

• y = 2x + 1

• y = 3x + 1

• y = 4x + 1

The distance of a point on ellipse $\frac{{\mathrm{x}}^2}{6} + \frac{{\mathrm{y}}^2}{2} = 1$ from its centre is 2. The eccentric angle of the point will be

• $\frac{\mathrm{\pi }}{4} \mathrm{or} \frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3} \mathrm{or} \frac{3\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{4} \mathrm{or} \frac{3\mathrm{\pi }}{4}$

• None of these

The distance between the foci of a hyperbola is 16 and its eccentncity is $\sqrt{2}$. Its equation will be

• x² - y² = 1

• x² - y² = 20

• x² - y² = 4

• x² - y² = 32

The point on x² = 2y, which is closest to the point (0, 5), will be

• $\left(2\sqrt{2}, 0\right)$

• (0, 0)

• (2, 2)

• None of these