Straight Lines(12th Grade > Mathematics ) Questions and answers for exam Preparation

12th Grade > Mathematics

STRAIGHT LINES MCQs

Straight Lines

Total Questions : 60 | Page 6 of 6 pages

Question 51. Let A (h, k), B (1, 1) and C (2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is 1, then the set of values which `k' can take is given by

{1, 3}
{0, 2}
{-1, 3}
{-3, -2}

Discuss Question

Answer: Option C. -> {-1, 3}
:
C
Since, A(h, k), B(1, 1) and C (2, 1) are the vertices of a right angled

Δ A B C

Let A (h, K), B (1, 1) And C (2, 1) Be The Vertices Of A Rig...

Now, area of

Δ A B C

= \frac{1}{2} | k - 1 | .1

\Rightarrow 1 = \frac{1}{2} | k - 1 |

\Rightarrow k - 1 = \pm 2

\Rightarrow k = - 1, 3

Question 52. If A(4, -3), B(3, -2) and C(2, 8) are the vertices of a triangle, then its centroid will be

(-3,3)
(3,3)
(3,1)
(1,3)

Discuss Question

Answer: Option C. -> (3,1)
:
C
Let the centroid of the triangle be (x, y).
The centroid of a triangle is given by

(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})

x = \frac{4 + 3 + 2}{3} = 3

y = \frac{- 3 - 2 + 8}{3} = 1

Question 53. O is the origin and A is the point (3,4). If a point P moves so that the line segment OP is always parallel to the line segment OA, then the equation to the locus of P is

4x - 3y = 0
4x + 3y = 0
3x + 4y = 0
3x - 4y = 0

Discuss Question

Answer: Option A. -> 4x - 3y = 0
:
A
Since OA and OP will be parallel only when O, A and P are collinear.
Therefore,

∣ ∣∣ ∣ \begin{matrix} 0 & 0 & 1 3 & 4 & 1 x & y & 1 \end{matrix} ∣ ∣∣ ∣

= 0

\Rightarrow

4x - 3y = 0.

Question 54. If the vertices of a triangle be (a, 1), (b, 3) and (4, c), then the centroid of the triangle will lie on x-axis, if

a + c = -4
a + b = -4
c = -4
b + c = -4

Discuss Question

Answer: Option C. -> c = -4
:
C
The point lies on axis of x, if y = 0.
Therefore,

\frac{1 + 3 + c}{3}

= 0

\Rightarrow

c = -4.

Question 55. If two vertices of a triangle are (6,4), (2,6) and its centroid is (4, 6), then the third vertex is

(4,8)
(8,4)
(6,4)
(0,0)

Discuss Question

Answer: Option A. -> (4,8)
:
A
Given:
Centroid =

(4, 6)

Vertices

(6, 4) & (2, 6)

Let the Co-ordinates of

C

(x_{3}, y_{3})

x_{1} = 6, x_{2} = 32, y_{1} = 4 & y_{2} = 6

Centroid

(4, 6) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})

\Rightarrow 4 = \frac{6 + 2 + x_{3}}{3}

and

6 = \frac{4 + 6 + y_{3}}{3}

\Rightarrow

x_{3} = 4

and

y_{3} = 8

∴

Third vertex is

(4, 8)

Question 56. If P = (1, 0), Q = (-1, 0) and R = (2, 0) are three given points, then the locus of a point S satisfying the relation

S Q^{2} + S R^{2} = 2 S P^{2}

A straight line parallel to x-axis
A circle through origin
A circle with centre at the origin
A straight line parallel to y-axis

Discuss Question

Answer: Option D. -> A straight line parallel to y-axis
:
D
Let S(x, y), then

(x + 1)^{2} + y^{2} + (x - 2)^{2} + y^{2} = 2 [(x - 1)^{2} + y^{2}]

\Rightarrow

2x +1 + 4 - 4x = - 4x + 2

\Rightarrow

x = -

\frac{3}{2}

Hence it is a straight line parallel to y-axis.

Question 57. If the vertices of a triangle be (a, b - c), (b, c - a) and (c, a - b), then the centroid of the triangle lies

At origin
On x-axis
On y-axis
(a+b+c,0)

Discuss Question

Answer: Option B. -> On x-axis
:
B
x =

\frac{a + b + c}{3}

, y =

\frac{b - c + c - a + a - b}{3}

= 0
Hence, centroid lies on x - axis.

Question 58. A point P moves so that its distance from the point (a, 0) is always equal to its distance from the line x + a = 0. The
locus of the point is

y2=4ax
x2=4ay
y2+4ax=0
x2+4ay=0

Discuss Question

Answer: Option A. -> y2=4ax
:
A

(x - a)^{2} + y^{2} = (x + a)^{2} \Rightarrow y^{2} = 4 a x

Note: This is also the definition of parabola

y^{2}

= 4ax.

Question 59. If h denote the A.M, k denote G.M of the intercepts made on axes by the lines passing through (1, 1) then (h, k) lies on

y2=2x
y2=4x
y=2x
x+y=2xy

Discuss Question

Answer: Option A. -> y2=2x
:
A
a = x - intercept, b =y - intercept

2 h = a + b, k^{2} = a b

\frac{x}{a} + \frac{y}{b} = 1

, substitute (1, 1)

\frac{1}{a} + \frac{1}{b} = 1

a + b = ab

2 h = k^{2} \Rightarrow y^{2} = 2 x

Question 60. If the orthocenter and circumcentre of a triangle are (0,0) and (3,6) respectively then the centroid of the triangle is

(1,2)
(2,4)
(23,43)
(13,23)

Discuss Question

Answer: Option B. -> (2,4)
:
B
In any triangle centroid divides the line joining orthocenter and circumcentre internally in the ratio 2 : 1.
Applying section formula to find the point which divides the line joining (0,0) in the ratio 2:1 , we get the coordinated of centroid equal to (2,4).