Find the value of λ such that lines x + 2 y = 3 , 3 x - y = 1 and λ x + y = 2 can not form a triangle.

Given that the lines x + 2 y - 3 = 0 , 3 x - y - 1 = 0 and λ x + y - 2 = 0 can not form a triangle, hence area of triangle formed by these lines equal to zero. So, we have two conditions for not forming the triangle: (i) If two lines are parallel to each other, they can not be a side of a triangle. (ii) If these lines are concurrent, they also can not form a triangle. Since, for parallel lines slope is same. Slope of lines x + 2 y - 3 = 0 , 3 x - y - 1 = 0 and λ x + y - 2 = 0 be - 1 2 , 3 and - λ respectively. ⇒ - λ = - 1 2 , 3 ⇒ λ = 1 2 , - 3 Since three lines are concurrent, if the point of intersection of two lines lies on the third line. So, solving the lines x + 2 y = 3 and 3 x - y = 1 ⇒ 3 3 - 2 y - y = 1 ⇒ y = 8 7 and x = 3 - 2 8 7 = 5 7 So, coordinates of point of intersection 5 7 , 8 7 should satisfy λ x + y - 2 = 0 ⇒ 5 7 λ + 8 7 - 2 = 0 ⇒ 5 λ + 8 - 14 = 0 ⇒ λ = 6 5 Hence, possible values of λ will be 6 5 , 1 2 and - 3 .

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 49

HARD

JEE Main/Advance

IMPORTANT

Earn 100

Find the equation of a straight line passing through the point $(4, 5)$ and equally inclined to the lines $3 x = 4 y + 7$ and $5 y = 12 x + 6 .$

Important Questions on Point and Straight Line

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 50

The line

x + 3 y - 2 = 0

bisects the angle between a pair of straight lines of which one has equation

x - 7 y + 5 = 0,

then find equation of other line.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 51

Find the value of

y - 5 = \frac{9}{7} (x - 4)

such that lines

x + 2 y = 3, 3 x - y = 1

and

λ x + y = 2

can not form a triangle.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 52

Find values of

λ

for which line

y = x + 1, y = λ x + 2

and

y = (λ^{2} + λ - 1) x + 3

are concurrent.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 53

Find the equation to the straight line passing through the point

(3, 2)

and the point of intersection of the lines

2 x + 3 y = 1

and

3 x - 4 y = 6

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 54

(ii)

The intersection of the lines

x + 2 y + 3 = 0

and

3 x + 4 y + 7 = 0

and perpendicular to the straight line

y - x = 8

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-1>Q 55

Find the locus of the circumcentre of a triangle whose two sides are along the coordinate axes and third side passes through the point of intersection of the lines