Mathematics · Ch 5 — Two Dimensional Analytical Geometry-II

Summary

5.9

Summary

Circle. Standard form $(x-h)^2+(y-k)^2=r^2$ (centre $(h,k)$ , radius $r$ ); general form $x^2+y^2+2gx+2fy+c=0$ (centre $(-g,-f)$ , radius $\sqrt{g^2+f^2-c}$ ); family through a line–circle intersection $S+\lambda L=0$ ; diameter form $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$ ; tangent at $(x_1,y_1)$ : $xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$ ; normal at $(x_1,y_1)$ : $xy_1-yx_1-g(y-y_1)-f(x-x_1)=0$ .

Conic definition. $SP=e\cdot PM$ (focus $S$ , directrix, eccentricity $e$ ): $e=1$ parabola, $0<e<1$ ellipse, $e>1$ hyperbola. General second-degree $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ ; identification checklist in §5.4.3 (circle/parabola/ellipse/hyperbola/point/empty-set/axes/pair-of-lines, by $A,B,C$ and the rest).

Table 1 — Tangent and normal.

Curve	Equation	Tangent	Normal
Circle	$x^2+y^2=a^2$	$xx_1+yy_1=a^2$ ; or $\frac{x\cos\theta}a+\frac{y\sin\theta}a=1$	$xy_1-yx_1=0$ ; or $\frac x{\sin\theta}-\frac y{\cos\theta}=0$
Parabola	$y^2=4ax$	$yy_1=2a(x+x_1)$ ; or $yt=x+at^2$	$xy_1+2ay=x_1y_1+2ay_1$ ; or $y+xt=2at+at^3$
Ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	$\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1$ ; or $\frac{x\cos\theta}a+\frac{y\sin\theta}b=1$	$\frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2$ ; or $\frac{ax}{\cos\theta}-\frac{by}{\sin\theta}=a^2-b^2$
Hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$\frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1$ ; or $\frac{x\sec\theta}a-\frac{y\tan\theta}b=1$	$\frac{a^2x}{x_1}+\frac{b^2y}{y_1}=a^2+b^2$ ; or $\frac{ax}{\sec\theta}+\frac{by}{\tan\theta}=a^2+b^2$

Table 2 — Tangency of $y=mx+c$ .

Conic	Condition	Point of contact	Tangent
Circle $x^2+y^2=a^2$	$c^2=a^2(1+m^2)$	$\left(\mp\frac{am}{\sqrt{1+m^2}},\pm\frac a{\sqrt{1+m^2}}\right)$	$y=mx\pm a\sqrt{1+m^2}$
Parabola $y^2=4ax$	$c=\frac am$	$\left(\frac{a}{m^2},\frac{2a}m\right)$	$y=mx+\frac am$

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