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Three Dimensional Geometry

Short Answer Type

61.

Find the angle subtended by the double ordinate of length 2a of the parabola y² = ax at its vertex.

62.

The equation to the pairs of opposite sides of a parallelogram are x²- 5x + 6 = 0 and y² - 6y + 5. Find the equations its diagonals.

Multiple Choice Questions

63.

A common tangent to 9x² - 16y² = 144 and x² + y² = 9 is

$y = \frac{3}{\sqrt{7}} x + \frac{15}{\sqrt{7}}$
$y = 3 \sqrt{\frac{2}{7}} x + \frac{15}{\sqrt{7}}$
$y = 2 \sqrt{\frac{3}{7}} x + 15 \sqrt{7}$
None of these

64.

The locus of a point P which moves such that 2PA = 3PB, where A(0, 0) and B(4,- 3) are points, is

5x² - 5y² - 72x + 54y + 225 = 0
5x² + 5y² - 72x + 54y + 225 = 0
5x² + 5y² + 72x - 54y + 225 = 0
5x² + 5y² - 72x - 54y - 225 = 0

65.

The points P is equidistant from A(1, 3), B (- 3, 5) and C(5, - 1), then PA is equal to

5
$5 \sqrt{5}$
25
$5 \sqrt{10}$

66.

The equation of the plane through the intersection of the planes x + y + z = 1 and 2x + 3y - z + 4 = 0 and parallel to X-axis, is

y - 3z + 6 = 0
3y - z + 6 = 0
y + 3z + 6 = 0
3y - 2z + 6 = 0

67.
If the direction cosines of two lines are connected by the equations l + m + n = 0, l² + m² - n² = 0, then the angle between the lines is

$\frac{π}{4}$

$\frac{π}{6}$

$\frac{π}{2}$

$\frac{π}{3}$

$\frac{π}{3}$

We have, l + m + n = 0 and l² + m² - n² = 0

$\Rightarrow l^{2} + m^{2} - {(- l - m)}^{2} = 0 \Rightarrow lm = 0 \Rightarrow l = 0 or m = 0$

When l = 0, then

$\Rightarrow I + m + n = 0 and l^{2} + m^{2} - n^{2} = 0 \Rightarrow m + n = 0 and m^{2} - n^{2} = 0 \Rightarrow m + n = 0$

$\Rightarrow m = - n ∴ \frac{l}{0} = \frac{m}{1} = \frac{n}{- 1} When m = 0, then l + m + n = 0 and l^{2} + m^{2} - n^{2} = 0 \Rightarrow l + n = 0 and l^{2} - n^{2} = 0 \Rightarrow l + n = 0 \Rightarrow l = - n ∴ \frac{l}{1} = \frac{m}{0} = \frac{n}{- 1}$

Thus, the direction ratios of two lines are proportional to 0, 1, - 1 and 1, 0, - 1

Let $θ$ be the angle between the lines.

$Then, \cos (θ) = \frac{1 \times 0 + 1 \times 0 + (- 1) (- 1)}{\sqrt{0 + 1 + 1} \sqrt{1 + 0 + 1}} = \frac{1}{2} \Rightarrow θ = \frac{π}{3}$

68.

The equation of the plane which contains the origin and the line of intersection of the planes r · a = d₁ and r · b = d₂, is

r . (d₁a + d₂b) = 0
r . (d₂a - d₁b) = 0
r . (d₂a + d₁b) = 0
r . (d₁a - d₂b) = 0

69.

If from a point P(a, b, c) perpendiculars PA and PB are drawn to YZ and ZX - planes, then the equation of the plane OAB is

bcx + cay + abz = 0
bcx + cay - abz = 0
- bcx + cay + abz = 0
bcx - cay + abz = 0

70.

If (2, 7, 3) is one end of a diameter of the sphere x² + y²+ z²- 6x - 12y - 2z + 20 = 0, then the coordinates of the other end of the diameter are

(- 2, 5, - 1)
(4, 5, 1)
(2, - 5, 1)
(4, 5. - 1)