Question

Assertion :If the area of the triangle formed by the lines $y=x,x+y=4$ and the lines through $P(h,k)$ parallel to y-axis is $k^2$, then $P$ lies on $x^2+y^2-4x+4=0$ Reason: Area of the triangle formed by the lines $y=x,x+y=4$ and the x-axis is equal to half the area of the triangle formed by the line $x+y=4$ and the coordinate axes.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion
Let the third side be $x=h$, then the vertices are $(2,2),(h,h)$ and $(h,4-h)$
$\Rightarrow k^2=\frac{1}{2}\left|\begin{array}{ccc}2& 2& 1\\ h& h& 1\\ h& 4-h& 1\end{array}\right|\Rightarrow 2k^2=\left|\begin{array}{ccc}2& 2& 1\\ h-2& h-2& 0\\ h-2& 2-h& 0\end{array}\right|\Rightarrow k^2+{(h-2)}^2=0$
So, $P(h,k)$ lies on $x^2+y^2-4x+4=0$
Hence assertion is true
Reason:
Area of the first triangle is $4$ and that of the second is $2$.
so it is true.