## Straight Lines Questions and Answers Part-3

1. The line x + y = 1 meets x-axis at A and y-axis at B. P is the mid-point of AB. $P_{1}$ is the foot of the perpendicular from P to OA; M1 is that from $P_{1}$ to OP; $P_{2}$ is that from $M_{1}$ to OA; $M_{2}$ is that from $P_{2}$ to OP; $P_{3}$ is that from $M_{2}$ to OA and so on. If $P_{n}$ denotes the nth foot of the perpendicular on OA from $M_{n-1}$   , then $OP_{n}$ =
a) 1/2
b) $1/2^{n}$
c) $1/2^{n/2}$
d) $1/\sqrt{2}$

Explanation: x + y = 1 meets x-axis at A(1, 0) and y-axis at B(0, 1)

2. The line x + y = a, meets the axis of x and y at A and B respectively. A triangle AMN is inscribed in the triangle OAB, O being the origin, with right angle at N. M and N lie respectively on OB and AB. If the area of the triangle AMN is 3/8 of the area of the triangle OAB, then AN/BN is equal to
a) 3
b) 1/3
c) 2
d) 1/2

Explanation:

3.The point (4, 1) undergoes the following transformation successively.
(i) reflection about the line y = x.
(ii) translation through a distance 2 units along the positive direction of x-axis.
(iii) rotation through an angle $\pi$/4 about the origin in the anticlockwise direction.
(iv) reflection about x = 0
The final position of the given point is
a) $\left(1/\sqrt{2},7/2\right)$
b) $\left(1/2,7/\sqrt{2}\right)$
c) $\left(1/\sqrt{2},7/\sqrt{2}\right)$
d) (1/2, 7/2)

Explanation: Let B, C, D, E be the positions of the given point A(4, 1) after the transformations (i), (ii), (iii) and (iv) successively

4. A line cuts the x-axis at A(7, 0) and the y-axis at B(0, – 5). A variable line PQ is drawn perpendicular to AB. Cutting the x-axis at P and the y-axis at Q. If AQ and BP intersect at R, the locus of R is
a) $x^{2}+y^{2}+7x-5y=0$
b) $x^{2}+y^{2}-7x+5y=0$
c) $5x-7y=35$
d) none of these

Explanation: Let P(a, 0) and Q(0, b).

5. Equation of the line which bisects the obtuse angle between the lines x – 2y + 4 = 0 and 4x – 3y + 2 = 0 is
a) $\left(4 + \sqrt{5}\right)x - \left(3 + 2 \sqrt{5}\right) y + 2 + 4 \sqrt{5} = 0$
b) $\left(4 - \sqrt{5}\right)x - \left(3 + 2 \sqrt{5}\right) y + 2 - 4 \sqrt{5} = 0$
c) $\left(4 - \sqrt{5}\right)x - \left(3 - 2 \sqrt{5}\right) y + 2 - 4 \sqrt{5} = 0$
d) $\left(4 + \sqrt{5}\right)x - \left(3 - 2 \sqrt{5}\right) y + 2 + 4 \sqrt{5} = 0$

Explanation: Let us form a triangle with sides x – 2y + 4 = 0, 4x – 3y + 2 = 0 and x = 0. Coordinates of the vertices of this triangle are A(8/5, 14/5), B(0, 2/3) and C(0, 2).

6. If the pairs of lines $x^{2}+2xy+ay^{2}=0$     and $ax^{2}+2xy+y^{2}=0$     have exactly one line in common then the joint equation of the other two lines is given by
a) $3x^{2}+8xy-3y^{2}=0$
b) $3x^{2}+10xy+3y^{2}=0$
c) $y^{2}+2xy-3x^{2}=0$
d) $x^{2}+2xy-3y^{2}=0$

Explanation: Let y = mx be a line common to the given pairs of lines, then

7. If the lines joining the origin to the intersection of the line y = mx + 2 and the curve x2 + y2 = 1 are at right angles, then
a) $m^{2}=1$
b) $m^{2}=3$
c) $m^{2}=7$
d) $2m^{2}=1$

Explanation: Joint equation of the lines joining the origin

8. Let PQR be a right angled isosceles triangle right angled at P (2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is
a) $3x^{2}-3y^{2}+8xy+20x+10y+25=0$
b) $3x^{2}-3y^{2}+8xy-20x-10y+25=0$
c) $3x^{2}-3y^{2}+8xy+10x+15y+20=0$
d) $3x^{2}-3y^{2}-8xy-10x-15y-20=0$

Explanation: Let the slopes of PQ and PR be m and

9. If $\theta$ is an angle between the lines given by the equation $6x^{2}+5xy-4y^{2}+7x+13y-3=0$
then equation of the line passing through the point of intersection of these lines and making an angle $\theta$ with the positive x-axis is
a) 2x + 11y + 13 = 0
b) 11x – 2y + 13 = 0
c) 2x – 11y + 2 = 0
d) 11x + 2y – 11 = 0

10. If one of the lines given by the equation $2x^{2}+axy+3y^{2}=0$     coincide with one of those given by $2x^{2}+bxy-3y^{2}=0$     and the other lines represented by them be perpendicular, then