Straight Line and Pair of Straight Lines 2 Question 6
6. The area of the triangle formed by the intersection of line parallel to $X$-axis and passing through $(h, k)$ with the lines $y=x$ and $x+y=2$ is $4 h^{2}$. Find the locus of point $P$.
(2005)
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Answer:
Correct Answer: 6. $2 x= \pm(y-1)$
Solution:
- Here, the triangle formed by a line parallel to $X$-axis passing through $P(h, k)$ and the straight line $y=x$ and $y=2-x$ could be as shown below:
Since, area of $\triangle A B C=4 h^{2}$
$\therefore \quad \frac{1}{2} A B \cdot A C=4 h^{2}$
where, $A B=\sqrt{2}|k-1|$ and $A C=\sqrt{2}(|k-1|)$
$$ \begin{aligned} & \Rightarrow & \frac{1}{2} \cdot 2(k-1)^{2} & =4 h^{2} \\ & \Rightarrow & 4 h^{2} & =(k-1)^{2} \\ & \Rightarrow & 2 h & = \pm(k-1) \end{aligned} $$
The locus of a point is $2 x= \pm(y-1)$.
