Equations of Normal in Different Forms
Category : JEE Main & Advanced
(1) Point form
| Equation of normals of all other standard parabolas at \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\] | |
| Equation of parabola | Normal at \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\] |
| \[{{y}^{2}}=\text{ }4ax\] | \[y-{{y}_{1}}=\frac{-{{y}_{1}}}{2a}(x-{{x}_{1}})\] |
| \[{{y}^{2}}=-4ax\] | \[y-{{y}_{1}}=\frac{{{y}_{1}}}{2a}(x-{{x}_{1}})\] |
| \[{{x}^{2}}=4ay\] | \[y-{{y}_{1}}=-\frac{2a}{{{x}_{1}}}(x-{{x}_{1}})\] |
| \[{{x}^{2}}=-4ay\] | \[y-{{y}_{1}}=\frac{2a}{{{x}_{1}}}(x-{{x}_{1}})\] |
(2) Parametric form
| Equations of normal of all other standard parabola at \[\mathbf{'t'}\] | ||
| Equations of parabolas | Parametric co-ordinates | Normals at \[\mathbf{'t'}\] |
| \[{{y}^{2}}=4ax\] | \[(a{{t}^{2}},\text{ }2at)\] | \[y+tx=2at+a{{t}^{3}}\] |
| \[{{y}^{2}}=-4ax\] | \[(-a{{t}^{2}},\,\,2at)\] | \[y-tx=2at+a{{t}^{3}}\] |
| \[{{x}^{2}}=4ay\] | \[(2at,\,\,a{{t}^{2}})\] | \[x+ty=2at+a{{t}^{3}}\] |
| \[{{x}^{2}}=-4ay\] | \[(2at,\,\,-a{{t}^{2}})\] | \[x-ty=2at+a{{t}^{3}}\] |
(3) Slope form
| Equations of normal, point of contact, and condition of normality in terms of slope (m) | |||
| Equations of parabola | Point of contact in terms of slope (m) | Equations of normal in terms of slope (m) | Condition of normality |
| \[{{y}^{2}}=4ax\] | \[(a{{m}^{2}},-2am)\] | \[y=mx-2am-a{{m}^{3}}\] | \[c=-2am-a{{m}^{3}}\] |
| \[{{y}^{2}}=-4ax\] | \[(-a{{m}^{2}},2am)\] | \[y=mx+2am+a{{m}^{3}}\] | \[c=2am+a{{m}^{3}}\] |
| \[{{x}^{2}}=4ay\] | \[\left( -\frac{2a}{m},\frac{a}{{{m}^{2}}} \right)\] | \[y=mx+2a+\frac{a}{{{m}^{2}}}\] | \[c=2a+\frac{a}{{{m}^{2}}}\] |
| \[{{x}^{2}}=-4ay\] | \[\left( \frac{2a}{m},-\frac{a}{{{m}^{2}}} \right)\] | \[y=mx-2a-\frac{a}{{{m}^{2}}}\] | \[c=-2a-\frac{a}{{{m}^{2}}}\] |
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