Visual SolutionPYQ 2024 · Jan Shift 2Tricky

Question

Let R be the interior region between the lines

3x - y + 1 = 0

and

x + 2y - 5 = 0

containing the origin. The set of all values of

a

, for which the points

(a^2, a+1)

lie in

R

, is:

(A)

(-3,-1) \cup \left(-\frac{1}{3}, 1\right)

(B)

(-3,0) \cup \left(\frac{1}{3}, 1\right)

(C)

(-3,0) \cup \left(\frac{2}{3}, 1\right)

(D)

(-3,-1) \cup \left(\frac{1}{3}, 1\right)

Solution Path

Same side conditions:

L_1

gives

a(3a-1)>0

L_2

gives

(a+3)(a-1)<0

. Intersection:

(-3,0) \cup (1/3, 1)

01Question Setup

1/4

Find the values of

a

for which the origin lies on the same side of two given lines.

a \in \;?

02Same Side as Origin

2/4

L_1

3a^2 - a > 0

gives

a \in (-\infty, 0) \cup (\tfrac{1}{3}, \infty)

L_2

a^2 + 2a - 3 < 0

gives

a \in (-3, 1)

L_1: a(3a-1) > 0

L_2: (a+3)(a-1) < 0

03IntersectionKEY INSIGHT

3/4

Intersect the two sets:

(-3, 0) \cup (\tfrac{1}{3}, 1)

a \in (-3, 0) \cup \left(\frac{1}{3}, 1\right)

04Final Answer

4/4

The intersection of both conditions gives the final answer.

\boxed{(-3, 0) \cup \left(\tfrac{1}{3}, 1\right)}

Concepts from this question1 concepts unlocked

★

EASY

Two points are on the same side of ax + by + c = 0 if they give the same sign when substituted.

(ax_1 + by_1 + c)(ax_2 + by_2 + c) > 0

Key for region problems where you need to check which side of a line a point lies on.

Region problemsInequality constraintsLocus

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