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A-line-has-equation-y-2x-7-and-a-curve-has-equation-y-x-2-4x-c-where-c-is-a-constant-Find-the-set-of-possible-values-of-c-for-which-the-line-does-not-intersect-the-curve-

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Question Number 25199 by tawa tawa last updated on 06/Dec/17
A line has equation y = 2x − 7 and a curve has equation y = x^2 − 4x + c, where c is a constant. Find the set of possible values of c for which the line does not intersect the curve.
$$\mathrm{A}\:\mathrm{line}\:\mathrm{has}\:\mathrm{equation}\:\mathrm{y}\:=\:\mathrm{2x}\:−\:\mathrm{7}\:\mathrm{and}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{has}\:\mathrm{equation}\:\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\mathrm{c}, \\ $$$$\mathrm{where}\:\mathrm{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{c}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{intersect}\:\mathrm{the}\:\mathrm{curve}. \\ $$
Commented by tawa tawa last updated on 06/Dec/17
options (a) c < 2 (b) c > 2 (c) c = 2 (d) c > 1/0.5
$$\mathrm{options} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{c}\:<\:\mathrm{2}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\mathrm{c}\:>\:\mathrm{2}\:\:\:\:\:\left(\mathrm{c}\right)\:\:\mathrm{c}\:=\:\mathrm{2}\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\:\:\:\mathrm{c}\:>\:\mathrm{1}/\mathrm{0}.\mathrm{5} \\ $$
Answered by Rasheed.Sindhi last updated on 06/Dec/17
y = 2x − 7 y = x^2 − 4x + c Solving above two simultaneously x^2 − 4x + c=2x−7 x^2 −6x+c+7=0 x=((6±(√(36−4c−28)))/2) x=((6±2(√(2−c)))/2)=3±(√(2−c)) x∈R⇒ c≤2 For x=3±(√(2−c)) where c≤2 the curve and the line intersect with eachother. So for c>2 , the line and curve will not intersect. Option (b) is correct.
$$\mathrm{y}\:=\:\mathrm{2x}\:−\:\mathrm{7} \\ $$$$\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\mathrm{c} \\ $$$$\mathrm{Solving}\:\mathrm{above}\:\mathrm{two}\:\mathrm{simultaneously} \\ $$$$\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\mathrm{c}=\mathrm{2x}−\mathrm{7} \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{6x}+\mathrm{c}+\mathrm{7}=\mathrm{0} \\ $$$$\mathrm{x}=\frac{\mathrm{6}\pm\sqrt{\mathrm{36}−\mathrm{4c}−\mathrm{28}}}{\mathrm{2}} \\ $$$$\mathrm{x}=\frac{\mathrm{6}\pm\mathrm{2}\sqrt{\mathrm{2}−\mathrm{c}}}{\mathrm{2}}=\mathrm{3}\pm\sqrt{\mathrm{2}−\mathrm{c}} \\ $$$$\mathrm{x}\in\mathbb{R}\Rightarrow\:\mathrm{c}\leqslant\mathrm{2} \\ $$$$\mathrm{For}\:\mathrm{x}=\mathrm{3}\pm\sqrt{\mathrm{2}−\mathrm{c}}\:\:\mathrm{where}\:\mathrm{c}\leqslant\mathrm{2}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{line}\:\mathrm{intersect}\:\mathrm{with}\:\mathrm{eachother}. \\ $$$$\mathrm{So}\:\mathrm{for}\:\:\mathrm{c}>\mathrm{2}\:,\:\mathrm{the}\:\mathrm{line}\:\mathrm{and}\:\mathrm{curve}\:\mathrm{will}\:\mathrm{not} \\ $$$$\mathrm{intersect}. \\ $$$$\mathrm{Option}\:\left(\mathrm{b}\right)\:\mathrm{is}\:\mathrm{correct}. \\ $$
Commented by tawa tawa last updated on 06/Dec/17
I really appreciate sir. God bless you sir.
$$\mathrm{I}\:\mathrm{really}\:\mathrm{appreciate}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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