Question

Assertion :If $x+ky=1$ and $x=a$ are the equations of the hypotenuse and a side of a right angled isosceles triangle then $k=\pm a$. Reason: Each side of a right angled isosceles triangle makes an angle $\pm \frac{\pi }{4}$ with the hypotenuse.

• 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 incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

Given sides

$x+ky=1$ ----(1)

and $x=a$ ---(2)

On comparing both eq with $y=mx+c$, we get

$m_1=-\frac{1}{k}$ and $m_2=1$

And the third side of the triangle is perpendicular to $x-k=0$

So, slope of third line be $m_2{m}_3=-1$

$m_3=-1$

The given triangle is isosceles triangle so the angle between line (1) and (2) is equal to the angle between line (1) and (3)

$\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{m_1-m_3}{1+m_1{m}_3}\right|$

$\left|\frac{-\frac{1}{k}-1}{1-\frac{1}{k}}\right|=\left|\frac{-\frac{1}{k}+1}{1+\frac{1}{k}}\right|$

$\left|\frac{-1-k}{k-1}\right|=\left|\frac{-1+k}{k+1}\right|$

$|-(1+k{)}^2|=\left|\right(k-1{)}^2|$

$|-1-k^2-2k|=|k^2+1-2k|$

$2k^2=|-2|$

$k^2=1$

$k=\pm 1$

Hence, the reason is correct.