SATHEE: Straight Line and Pair of Straight Lines 1 Question 53

Straight Line and Pair of Straight Lines 1 Question 53

53.

For points $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ of the coordinate plane, a new distance $d (P, Q)$ is defined by $d (P, Q) = | x_{1} - x_{2} | + | y_{1} - y_{2} |$ .

Let $O = (0, 0)$ and $A = (3, 2)$ . Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

$(2000, 10 M)$

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Solution:

$d : (P, Q) = | x_{1} - x_{2} | + | y_{1} - y_{2} |$ .

It is new method of representing distance between two points $P$ and $Q$ and in future very important in coordinate geometry.

Now, let $P (x, y)$ be any pont in the first quadrant. We have

$d (P, 0)$ $= | x - 0 | + | y - 0 | = | x | + | y | = x + y$

$[∵ x, y > 0]$

$d (P, A)$ $= | X - 3 | + | Y - 2 |$ [given]

$d (P, 0)$ $= d (P, A)$ [given]

$\Rightarrow x + y$ $= | x - 3 | + | y - 2 |$ $\dots (i)$

Case I When $0 < x < 3, 0 < y < 2$

In this case, Eq. (i) becomes

$x + y = 3 - x + 2 - y$

$\begin{aligned} \Rightarrow & 2 x + 2 y & = 5 \\ or & x + y & = 5 / 2 \end{aligned}$

Case II When $0 < x < 3, y \geq 2$

Now, Eq. (i) becomes

$\begin{aligned} x + y = 3 - x + y - 2 \\ \Rightarrow 2 x = 1 \\ \Rightarrow x = 1 / 2 \end{aligned}$

Case III When $x \geq 3, 0 < y < 2$

Now, Eq. (i) becomes

$\begin{aligned} x + y = x - 3 + 2 - y \\ \Rightarrow 2 y = - 1 or y = - 1 / 2 \end{aligned}$

Hence, no solution.

Case IV When $x \geq 3, y \geq 2$

In this case, case I changes to

$x + y = x - 3 + y - 2 \Rightarrow 0 = - 5$

which is not possible.

Hence, the solution set is

$(x, y) ∣ x = 12, y \geq 2 \cup (x, y) ∣$

$x + y = 5 / 2, 0 < x < 3, 0 < y > 2$

The graph is given in adjoining figure.

Straight Line and Pair of Straight Lines 1 Question 53

53.

Solution:

Engineering Preparation

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NCERT Exercise Video Solution

Straight Line and Pair of Straight Lines 1 Question 53

53.

Solution:

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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