SATHEE: Straight Line and Pair of Straight Lines 2 Question 8

Straight Line and Pair of Straight Lines 2 Question 8

8. Lines $L_{1} \equiv a x + b y + c = 0$ and $L_{2} \equiv l x + m y + n = 0$ intersect at the point $P$ and makes an angle $θ$ with each other. Find the equation of a line $L$ different from $L_{2}$ which passes through $P$ and makes the same angle $θ$ with $L_{1} \cdot y$

$(1988, 5 M)$

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Answer:

Correct Answer: 8. $2 (a l + b m) (a x + b y + c) - (a^{2} + b^{2}) (l x + m y + n) = 0$

Solution:

Since, the required line $L$ passes through the intersection of $L_{1} = 0$ and $L_{2} = 0$ .

So, the equation of the required line $L$ is

$L_{1} + λ L_{2} = 0$

i.e. $(a x + b y + c) + λ (l x + m y + n) = 0$

where, $λ$ is a parameter.

Since, $L_{1}$ is the angle bisector of $L = 0$ and $L_{2} = 0$ .

$∴$ Any point $A (x_{1}, y_{1})$ on $L_{1}$ is equidistant from $L_{1} = 0$ and $L_{2} = 0$ .

$\begin{aligned} \Rightarrow \frac{| l x_{1} + m y_{1} + n |}{\sqrt{l^{2} + m^{2}}} \\ = \frac{| (a x_{1} + b y_{1} + c) + λ (l x_{1} + m y_{1} + n) |}{\sqrt{(a + λ l)^{2} + (b + λ m)^{2}}} \end{aligned}$

But, $A (x_{1}, y_{1})$ lies on $L_{1}$ . So, it must satisfy the equation of $L_{1}$ , ie, $a x_{1} + b y_{1} + c_{1} = 0$ .

On substituting $a x_{1} + b y_{1} + c = 0$ in Eq. (ii), we get

$\begin{aligned} \frac{| l x_{1} + m y_{1} + n |}{\sqrt{l^{2} + m^{2}}} & = \frac{| 0 + λ (l x_{1} + m y_{1} + n) |}{\sqrt{(a + λ l)^{2} + (b + λ m)^{2}}} \\ \Rightarrow & λ^{2} (l^{2} + m^{2}) & = (a + λ l)^{2} + (b + λ m)^{2} \\ ∴ & λ & = - \frac{(a^{2} + b^{2})}{2 (a l + b m)} \end{aligned}$

On substituting the value of $λ$ in Eq. (i), we get

$(a x + b y + c) - \frac{(a^{2} + b^{2})}{2 (a l + b m)} (l x + m y + n) = 0$

$\Rightarrow 2 (a l + b m) (a x + b y + c) - (a^{2} + b^{2}) (l x + m y + n) = 0$

which is the required equation of line $L$ .

Straight Line and Pair of Straight Lines 2 Question 8

8. Lines $L_{1} \equiv a x + b y + c = 0$ and $L_{2} \equiv l x + m y + n = 0$ intersect at the point $P$ and makes an angle $θ$ with each other. Find the equation of a line $L$ different from $L_{2}$ which passes through $P$ and makes the same angle $θ$ with $L_{1} \cdot y$

Answer:

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Straight Line and Pair of Straight Lines 2 Question 8

8. Lines L1≡ax+by+c=0 and L2≡lx+my+n=0 intersect at the point P and makes an angle θ with each other. Find the equation of a line L different from L2 which passes through P and makes the same angle θ with L1⋅y

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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8. Lines $L_{1} \equiv a x + b y + c = 0$ and $L_{2} \equiv l x + m y + n = 0$ intersect at the point $P$ and makes an angle $θ$ with each other. Find the equation of a line $L$ different from $L_{2}$ which passes through $P$ and makes the same angle $θ$ with $L_{1} \cdot y$