SATHEE: Straight Line and Pair of Straight Lines 1 Question 54

Straight Line and Pair of Straight Lines 1 Question 54

54. A rectangle $P Q R S$ has its side $P Q$ parallel to the line $y = m x$ and vertices $P Q$ and $S$ on the lines $y = a, x = b$ and $x = - b$ , respectively. Find the locus of the vertex $R$ .

$(1996, 2 M)$

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Solution:

Let the coordinates of $Q$ be $(b, α)$ and that of $S$ be $(- b, β)$ . Suppose, $P R$ and $S Q$ meet in $G$ . Since, $G$ is mid point of $S Q$ , its $x$ -coordinate must be 0 . Let the coordinates of $R$ be $(h, k)$ .

Since, $G$ is mid point of $P R$ , the $x$ -coordinate of $P$ must be $- h$ and as $P$ lies on the line $y = a$ , the coordinates of $P$ are $(- h, a)$ . Since, $P Q$ is parallel to $y = m x$ , slope of $P Q = m$

$\Rightarrow \frac{α - a}{b + h} = m$

Again,

$Slope of R Q = - \frac{1}{m} \Rightarrow \frac{k - α}{h - b} = - \frac{1}{m}$

From Eq. (i), we get

$\begin{aligned} α - a & = m (b + h) \\ α & = a + m (b + h) \end{aligned}$

and from Eq. (ii), we get

$\begin{aligned} k - α & = - \frac{1}{m} (h - b) \\ \Rightarrow α & = k + \frac{1}{m} (h - b) \end{aligned}$

From Eqs. (iii) and (iv), we get

$\begin{aligned} a + m (b + h) = k + \frac{1}{m} (h - b) \\ \Rightarrow & a m + m^{2} (b + h) = k m + (h - b) \\ \Rightarrow & (m^{2} - 1) h - m k + b (m^{2} + 1) + a m = 0 \end{aligned}$

Hence, the locus of vertex is

$(m^{2} - 1) x - m y + b (m^{2} + 1) + a m = 0$

Straight Line and Pair of Straight Lines 1 Question 54

54. A rectangle $P Q R S$ has its side $P Q$ parallel to the line $y = m x$ and vertices $P Q$ and $S$ on the lines $y = a, x = b$ and $x = - b$ , respectively. Find the locus of the vertex $R$ .

Solution:

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Straight Line and Pair of Straight Lines 1 Question 54

54. A rectangle PQRS has its side PQ parallel to the line y=mx and vertices PQ and S on the lines y=a,x=b and x=−b, respectively. Find the locus of the vertex R.

Solution:

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Important Resources

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NCERT Exercise Video Solution

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54. A rectangle $P Q R S$ has its side $P Q$ parallel to the line $y = m x$ and vertices $P Q$ and $S$ on the lines $y = a, x = b$ and $x = - b$ , respectively. Find the locus of the vertex $R$ .