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Straight Lines and Pair of Straight Lines question

2006 · Shift 0 · Q77

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Straight Lines and Pair of Straight Lines
2006 · Shift 0 · Q77

Straight Lines and Pair of Straight Lines question

2006 · Shift 0 · Q77

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1

If

\left( {a,{a^2}} \right)

falls inside the angle made by the lines

y = {x \over 2},x \gt 0

and

y = 3x,x \gt 0,

then a belong to :

A
$\left( {0,{1 \over 2}} \right)$
B
$\left( {3,\infty } \right)$
C
$\left( {{1 \over 2},3} \right)$
D
$\left( {-3,-{1 \over 2}} \right)$

View written solutionFree

Correct answer: C

Interpret the angle region

The lines are $y=\frac{x}{2},\; x>0$ and $y=3x,\; x>0.$

Since both rays are for $x>0$ , they form an angle in the first quadrant between the two lines.

So a point $(x,y)$ lies inside this angle iff:

$x>0$ ,
it is above $y=\dfrac{x}{2}$ ,
it is below $y=3x$ .

Thus the condition is $\frac{x}{2}<y<3x, \quad x>0.$

Substitute the point

Given point is $(a,a^2).$

So we put $x=a,\quad y=a^2.$

Hence the conditions become: $a>0,$ $\frac{a}{2}<a^2<3a.$

Solve the inequalities

Since $a>0$ , we can divide throughout by $a$ safely.

From $\frac{a}{2}<a^2$ we get $\frac12<a.$

From $a^2<3a$ we get $a<3.$

Combining with $a>0$ : $\frac12<a<3.$

Match with the options

Thus, $a\in \left(\frac12,3\right).$

This corresponds to Option C.

Comparison with stored answer

Stored correct answer: C

Our derived answer: C

So the answer agrees with the stored correct answer.

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