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Question: Find the value of x so that the inclination of the line joining the points (x, -3) and (2, 5) is \[{...

Find the value of x so that the inclination of the line joining the points (x, -3) and (2, 5) is 135∘{{135}^{\circ }}.

Explanation

Solution

We know that the slope of a line joining the two points (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and (x2,y2)\left( {{x}_{2}},{{y}_{2}} \right) is equal to the tangent of the angle made by the line with x-axis in anticlockwise direction given by as follows:
slope=tan⁡θ=y2−y1x2−x1slope=\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

Complete step-by-step solution:
We have been given a line joining the points (x, -3) and (2, 5) which makes an angle of 135∘{{135}^{\circ }} with the x-axis.
We know that the slope of a line joining the two points (x1,y1)\left( {{x}_{1}},{{y}_{1}} \right) and (x2,y2)\left( {{x}_{2}},{{y}_{2}} \right) is equal to the tangent of the angle made by the line with x-axis in anticlockwise direction given by as follows:
slope=tan⁡θ=y2−y1x2−x1slope=\tan \theta =\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

So we have θ=135∘,x1=2,x2=x,y1=5,y2=−3\theta ={{135}^{\circ }},{{x}_{1}}=2,{{x}_{2}}=x,{{y}_{1}}=5,{{y}_{2}}=-3
⇒tan⁡135∘=−3−5x−2\Rightarrow \tan {{135}^{\circ }}=\dfrac{-3-5}{x-2}
Since we know that tan⁡135∘=tan⁡(90∘+45∘)=−cot⁡45∘=−1\tan {{135}^{\circ }}=\tan \left( {{90}^{\circ }}+{{45}^{\circ }} \right)=-\cot {{45}^{\circ }}=-1 as in second quadrant tangent function is negative.

& \Rightarrow -1=\dfrac{-3-5}{x-2} \\\ & \Rightarrow -1=\dfrac{-8}{x-2} \\\ \end{aligned}$$ On cross multiplication, we get as follows: $$\Rightarrow -x+2=-8$$ On adding - 2 to both the sides of the equation, we get as follows: $$\begin{aligned} & \Rightarrow -x-2+2=-8-2 \\\ & \Rightarrow -x=-10 \\\ & \Rightarrow x=10 \\\ \end{aligned}$$ **Therefore the value of x is equal to 10.** **Note:** Use the value of $$\tan {{135}^{\circ }}$$ very carefully as sometimes we used the value of $$\tan {{135}^{\circ }}$$ as equal to 1 which is which is wrong and thus gives the wrong answer. Also, substitute the values of $${{x}_{1}},{{x}_{2}},{{y}_{1}},{{y}_{2}}$$ carefully in the formula of slope of the line. Also, remember that in $$\tan \theta $$, $$\theta $$ is the angle between the line and the x-axis in an anticlockwise direction with the x-axis.