# Let $\dpi{100} A(h,k),B(1,1)\; and\; C(2,1)$ be the vertices of a right angled triangle with $\dpi{100} AC$ as its hypotenuse. If the area of the triangle is 1 square unit, then the set of values which $\dpi{100} ' k'$ can take is given by Option 1) {–1, 3} Option 2) {–3, –2} Option 3) {1, 3} Option 4) {0, 2}

S Sabhrant Ambastha

As we learnt in

Distance formula -

The distance between the point $A\left ( x_{1},y_{1} \right )\: and \: B\left ( x_{2},y_{2} \right )$

is $\sqrt{\left ( x_{1} -x_{2}\right )^{2}+\left ( y_{1} -y_{2}\right )^{2}}$

- wherein

$AC^{2}=AB^{2}+BC^{2}$

$(h-2)^{2}+(k-1)^{2}=(h+1)^{2}+(k-1)^{2}+1^{2}$

$4-4h=1-2h+1$

$2h=2\:\:\Rightarrow h=1$

Also,  $\frac{1}{2}AB.BC=1$

$\frac{1}{2}\times AB \times 1=1$

$AB=2=\sqrt{(h-1)^{2}+(k-1)^{2}}$

$(k-1)^{2}=2^{2}$

$k-1=2\:\:\:\:or\:\:\:\:k=-1=-2$

$k=3\:\:\:\:or\:\:\:\:k=-1$

Option 1)

{–1, 3}

This option is correct.

Option 2)

{–3, –2}

This option is incorrect.

Option 3)

{1, 3}

This option is incorrect.

Option 4)

{0, 2}

This option is incorrect.

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