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