SATHEE: Straight Line and Pair of Straight Lines 1 Question 56

Straight Line and Pair of Straight Lines 1 Question 56

56.

Determine all values of $α$ for which the point $(α, α^{2})$ lies inside the triangles formed by the lines $2 x + 3 y - 1 = 0$ ,

$x + 2 y - 3 = 0, 5 x - 6 y - 1 = 0$

(1992, 6M)

Show Answer

Answer:

Correct Answer: 56. ( $- \frac{3}{2} < α < - 1 \cup \frac{1}{2} < α < 1$ )

Solution:

Given lines are $2 x + 3 y - 1 = 0$ …….(i)

$x + 2 y - 3 = 0$ …….(ii)

$5 x - 6 y - 1 = 0$ …….(iii)

On solving Eqs. (i), (ii) and (iii), we get the vertices of a triangle are $A (- 7, 5), B (\frac{1}{3}, \frac{1}{9})$ and $C (\frac{5}{4}, \frac{7}{8})$ .

Let $P (α, α^{2})$ be a point inside the $△ A B C$ . Since, $A$ and $P$ are on the same side of $5 x - 6 y - 1 = 0$ , both $5 (- 7) - 6 (5) - 1$ and $5 α - 6 α^{2} - 1$ must have the same sign, therefore

$\begin{array}{cc} 5 α - 6 α^{2} - 1 < 0 \\ \Rightarrow & 6 α^{2} - 5 α + 1 > 0 \\ \Rightarrow & (3 α - 1) (2 α - 1) > 0 \\ α < \frac{1}{3} or α > \frac{1}{2} \dots \dots . (i v) \end{array}$

Also, since $P (α, α^{2})$ and $C (\frac{5}{4}, \frac{7}{8})$ lie on the same side of $2 x + 3 y - 1 = 0$ , therefore both $2 (\frac{5}{4}) + 3 (\frac{7}{8}) - 1$ and $2 α + 3 α^{2} - 1$ must have the same sign.

Therefore,

$2 α + 3 α^{2} - 1 > 0$

$\begin{aligned} \Rightarrow & (α + 1) α - \frac{1}{3} > 0 \\ \Rightarrow & α < - 1 \cup α > 1 / 3 \dots \dots . (v) \end{aligned}$

and lastly $(\frac{1}{3}, \frac{1}{9})$ and $P (α, α^{2})$ lie on the same side of the line therefore, $\frac{1}{3} + 2 (\frac{1}{9}) - 3$ and $α + 2 α^{2} - 3$ must have the same sign.

Therefore, $2 α^{2} + α - 3 < 0$

$\begin{array}{ll} \Rightarrow & 2 α (α - 1) + 3 (α - 1) < 0 \\ \Rightarrow & (2 α + 3) (α - 1) < 0 \Rightarrow - \frac{2}{3} < α < 1 \end{array}$

On solving Eqs. (i), (ii) and (iii), we get the common answer is $- \frac{3}{2} < α < - 1 \cup \frac{1}{2} < α < 1$ .

Straight Line and Pair of Straight Lines 1 Question 56

56.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Straight Line and Pair of Straight Lines 1 Question 56

56.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu