SATHEE: Straight Line and Pair of Straight Lines 5 Question 1

Straight Line and Pair of Straight Lines 5 Question 1

1.

Let $P Q R$ be a right angled isosceles triangle, right angled at $P (2, 1)$ . If the equation of the line $Q R$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

$(1999, 2 M)$

(a) $3 x^{2} - 3 y^{2} + 8 x y + 20 x + 10 y + 25 = 0$

(b) $3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0$

(d) $3 x^{2} - 3 y^{2} - 8 x y - 10 x - 15 y - 20 = 0$

Show Answer

Answer:

Correct Answer: 1. (b)

Solution:

Let $S$ be the mid-point of $Q R$ and given $△ P Q R$ is an isosceles.

Therefore, $P S ⊥ Q R$ and $S$ is mid-point of hypotenuse, therefore $S$ is equidistant from $P, Q, R$ .

$\begin{array}{lc} ∴ & P S = Q S = R S \\ Since, & ∠ P = 90^{\circ} and ∠ Q = ∠ R \\ But & ∠ P + ∠ Q + ∠ R = 180^{\circ} \end{array}$

Now, slope of $Q R$ is -2 .

[given]

But $Q R ⊥ P S$ .

$∴$ Slope of $P S$ is $1 / 2$ .

Let $m$ be the slope of $P Q$ .

$∴ \tan (\pm 45^{\circ}) = \frac{m - 1 / 2}{1 - m (- 1 / 2)}$

$\Rightarrow \pm 1 = \frac{2 m - 1}{2 + m}$

$\Rightarrow m = 3, - 1 / 3$

$∴$ Equations of $P Q$ and $P R$ are

$\begin{aligned} y - 1 = 3 (x - 2) \\ and y - 1 = - \frac{1}{3} (x - 2) \\ or 3 (y - 1) + (x - 2) = 0 \end{aligned}$

Therefore, joint equation of $P Q$ and $P R$ is

$[3 (x - 2) - (y - 1)] [(x - 2) + 3 (y - 1)] = 0$

$\Rightarrow 3 (x - 2)^{2} - 3 (y - 1)^{2} + 8 (x - 2) (y - 1) = 0$

$\Rightarrow 3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0$

Straight Line and Pair of Straight Lines 5 Question 1

1.

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 5 Question 1

1.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu