Straight Lines Questions and Answers Part-6

1.Equation of a line which is parallel to the line common to the pair of lines given by $6x^2-xy-12y^2=0$     and $15x^2+14xy-8y^2=0$      and at a distance 7 from it is
a) 3x + 4y = 35
b) 5x – 2y = 7
c) 3x + 4y = – 35
d) Both a and c

Answer: d
Explanation:

2. The lines joining the origin to the point of intersection of $3x^2+\lambda xy-4x+1=0$      and 2x + y – 1 = 0 are at right angles for
a) $\lambda=-4$
b) $\lambda=4$
c) $\lambda=7$
d) All of the Above

Answer: d
Explanation: Equation of the lines joining the origin to the points of intersection of the given lines is

3. If $x^2+2h xy+y^2=0$      represents the equations of the straight lines through the origin which make an angle $\alpha$ with the straight line y + x = 0, then
a) $\sec 2\alpha=h$
b) $\cos \alpha=\sqrt{\frac{1+h}{2h}}$
c) $\cot \alpha=\sqrt{\frac{h+1}{h-1}}$
d) All of the Above

Answer: d
Explanation: Let equation of the lines given by x2 + 2hxy + y2 = 0 be y = m1x and y = m2x. Since these make an angle $\alpha$ with y + x = 0 whose slope is –1,

4. If the lines 2(sin a + sin b)x – 2(sin a – b)y = 3 and 2(cos a + cos b)x + 2cos(a – b)y = 5 are perpendicular, then sin 2a + sin 2b is equal to
a) sin (a – b) – 2sin(a + b)
b) sin 2(a – b) – 2sin(a + b)
c) 2sin (a – b) – sin(a + b)
d) sin 2(a – b) – sin(a + b)

Answer: b
Explanation: Equation of the bisectors of the angles between the given lines is

5. A(0, 0), B(2, 0), C(2, 2), D(0, 2) are the vertices of a square ABCD. The square is rotated through an angle of 30° in the anticlockwise direction so that AB makes an angle of 30° with the positive direction of x-axis. Equation of the diagonal BD in the new position is
a) $\left( \sqrt{3} +1\right)x +\left( \sqrt{2} -1\right) y = 3$
b) $\left( 2-\sqrt{3} \right)x +y=2\left( \sqrt{3} -1\right)$
c) $\left( \sqrt{3}-1 \right)x +y=2-\sqrt{3}$
d) $\left( \sqrt{2}-1 \right)x +\left( \sqrt{3}+1\right) y=3$

Answer: b
Explanation: Let APQR be the new position of the square.

6. If the lines x + 2ay + a = 0, x + 3by + b = 0 and x + 4cy + c = 0 are concurrent, then a, b, c are in
a) A.P
b) G.P
c) H.P.
d) none of these

Answer: c
Explanation:

7. If the lines 2 (sin a + sin b) x - 2 sin (a - b) y = 3 and 2 (cos a + cos b) x + 2 cos (a - b)y = 5 are perpendicular, then sin 2a + sin 2b is equal to
a) sin (a - b) - 2 sin (a + b)
b) sin 2 (a - b) - 2 sin (a + b)
c) 2 sin (a - b) - sin (a + b)
d) sin 2 (a - b) - sin (a + b)

Answer: b
Explanation:

8. If $p_{1},p_{2}$   denote the lengths of the perpendiculars from the origin on the lines $x\sec\alpha+$   y cosec $\alpha=2a$   and $x\cos\alpha+y\sin\alpha=a\cos 2\alpha$      respectively, then $\left(\frac{p_{1}}{p_{2}}+\frac{p_{2}}{p_{1}}\right)^2$   is equal to
a) $4\sin^2 4\alpha$
b) $4\cos^2 4\alpha$
c) 4 cosec2 $4\alpha$
d) $4\sec^2 4\alpha$

Answer: c
Explanation:

9. The locus of the point of intersection of the lines $x\sin \theta+\left(1-\cos\theta\right)y=a\sin\theta$
and $x\sin \theta-\left(1+\cos\theta\right)y+a\sin\theta=0$
is
a) $x^2-y^2=a^2$
b) $x^2+y^2=a^2$
c) $y^2=ax$
d) none of these

Answer: b
Explanation:

10. The straight lines 4x - 3y - 5 = 0, x - 2y - 10 = 0, 7x + y - 40 = 0 and x + 3y + 10 = 0 form the sides of a
a) quadrilateral
b) cyclic quadrilateral
c) rectangle
d) parallelogram

Answer: b
Explanation: