Question Number 50977 by peter frank last updated on 22/Dec/18
$${Find}\:{interms}\:{of}\:\:{a},{b}\:{the} \\ $$$${value}\:{of}\:{c}\:{which}\:{makes} \\ $$$${the}\:{line}\:{y}={mx}+{c} \\ $$$${a}\:{tangent}\:{to}\:{the}\:{parabola} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{ax}.{also}\:{obtain}\:{the}\: \\ $$$${coordinate}\:{of}\:{the}\:{point}\:{of} \\ $$$${contact} \\ $$$$\left.{b}\right)\:{find}\:{the}\:{equation}\:{of}\: \\ $$$${tangent}\:\frac{{x}^{\mathrm{2}} }{\mathrm{4}}+\frac{{y}^{\mathrm{2}} }{\mathrm{9}}=\mathrm{1}\:{with} \\ $$$${gradient}\:\mathrm{2} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18
$$\left({mx}+{c}\right)^{\mathrm{2}} =\mathrm{4}{ax} \\ $$$${m}^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{2}{mcx}+{c}^{\mathrm{2}} =\mathrm{4}{ax} \\ $$$${x}^{\mathrm{2}} \left({m}^{\mathrm{2}} \right)+{x}\left(\mathrm{2}{mc}−\mathrm{4}{a}\right)+{c}^{\mathrm{2}} =\mathrm{0} \\ $$$${roots}\:{are}\:{equal}\:\:{B}^{\mathrm{2}} =\mathrm{4}{AC} \\ $$$$\left(\mathrm{2}{mc}−\mathrm{4}{a}\right)^{\mathrm{2}} =\mathrm{4}{m}^{\mathrm{2}} {c}^{\mathrm{2}} \\ $$$$\mathrm{4}{m}^{\mathrm{2}} {c}^{\mathrm{2}} −\mathrm{16}{amc}+\mathrm{16}{a}^{\mathrm{2}} =\mathrm{4}{m}^{\mathrm{2}} {c}^{\mathrm{2}} \\ $$$$−\mathrm{16}{amc}=−\mathrm{16}{a}^{\mathrm{2}} \\ $$$${c}=\frac{{a}}{{m}} \\ $$
Commented by peter frank last updated on 23/Dec/18
$${thank}\:{you}\:{sir}. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18
$$\left.{b}\right){tanngent}\: \\ $$$${y}={mx}+\sqrt{{a}^{\mathrm{2}} {m}^{\mathrm{2}} +{b}^{\mathrm{2}} }\: \\ $$$${y}=\mathrm{2}{x}+\sqrt{\mathrm{16}+\mathrm{9}}\: \\ $$$${y}=\mathrm{2}{x}\pm\mathrm{5} \\ $$
Commented by peter frank last updated on 23/Dec/18
$${thank}\:{you} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 23/Dec/18
$${most}\:{welcome}… \\ $$
Answered by peter frank last updated on 23/Dec/18
$$\left.{b}\right)\:{a}^{\mathrm{2}} =\mathrm{4}\:\:\:\:{b}^{\mathrm{2}} =\mathrm{9}\:\:\:\:{m}=\mathrm{2} \\ $$$${y}={mx}+{c} \\ $$$${c}^{\mathrm{2}} ={b}^{\mathrm{2}} +{a}^{\mathrm{2}} {m}^{\mathrm{2}} \\ $$$${c}=\pm\mathrm{5} \\ $$$${y}=\mathrm{2}{x}\pm\mathrm{5} \\ $$$$ \\ $$