Solve {l}{x^2/a^2+y^2/b^2=1}{y=k(x-1)}

$$\left\{ \begin{array} { l } { \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 } \\ { y = k ( x - 1 ) } \end{array} \right.$$

Solve for x, y

$$x=\frac{a\left(-b\sqrt{b^{2}-k^{2}+\left(ak\right)^{2}}+ak^{2}\right)}{b^{2}+\left(ak\right)^{2}}\text{, }y=-\frac{bk\left(a\sqrt{b^{2}-k^{2}+\left(ak\right)^{2}}+b\right)}{b^{2}+\left(ak\right)^{2}}$$

$$x=\frac{a\left(b\sqrt{b^{2}-k^{2}+\left(ak\right)^{2}}+ak^{2}\right)}{b^{2}+\left(ak\right)^{2}}\text{, }y=\frac{bk\left(a\sqrt{b^{2}-k^{2}+\left(ak\right)^{2}}-b\right)}{b^{2}+\left(ak\right)^{2}}\text{, }\left(b\neq 0\text{ and }k\neq 0\text{ and }a\neq 0\text{ and }|a|\geq \frac{\sqrt{\left(k-b\right)\left(b+k\right)}}{|k|}\text{ and }b\leq -k\text{ and }b\geq k\right)\text{ or }\left(b\neq 0\text{ and }k\neq 0\text{ and }a\neq 0\text{ and }|a|\geq \frac{\sqrt{\left(k-b\right)\left(b+k\right)}}{|k|}\text{ and }b\leq k\text{ and }b\geq -k\right)\text{ or }\left(a\neq 0\text{ and }|b|>|k|\text{ and }b\neq 0\right)\text{ or }\left(a\neq 0\text{ and }k\neq 0\text{ and }|b|=|k|\right)\text{ or }\left(b\neq 0\text{ and }a\neq 0\text{ and }|a|\geq \frac{\sqrt{\left(k-b\right)\left(b+k\right)}}{|k|}\text{ and }k\neq 0\text{ and }b=|k|\right)$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Multiply both sides of the equation by a^{2}b^{2}, the least common multiple of a^{2},b^{2}.

$$b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$$

Consider the second equation. Use the distributive property to multiply k by x-1.

$$y=kx-k$$

Substitute kx-k for y in the other equation, b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}.

$$b^{2}x^{2}+a^{2}\left(kx-k\right)^{2}=a^{2}b^{2}$$

Square kx-k.

$$b^{2}x^{2}+a^{2}\left(k^{2}x^{2}+2\left(-k\right)kx+\left(-k\right)^{2}\right)=a^{2}b^{2}$$

Multiply a^{2} times k^{2}x^{2}+2\left(-k\right)kx+\left(-k\right)^{2}.

$$b^{2}x^{2}+a^{2}k^{2}x^{2}+2\left(-k\right)ka^{2}x+\left(-k\right)^{2}a^{2}=a^{2}b^{2}$$

Add b^{2}x^{2} to a^{2}k^{2}x^{2}.

$$\left(a^{2}k^{2}+b^{2}\right)x^{2}+2\left(-k\right)ka^{2}x+\left(-k\right)^{2}a^{2}=a^{2}b^{2}$$

Subtract a^{2}b^{2} from both sides of the equation.

$$\left(a^{2}k^{2}+b^{2}\right)x^{2}+2\left(-k\right)ka^{2}x+\left(-k\right)^{2}a^{2}-a^{2}b^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute b^{2}+a^{2}k^{2} for a, a^{2}\times 2k\left(-k\right) for b, and -\left(k+b\right)\left(-k+b\right)a^{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2\left(-k\right)ka^{2}±\sqrt{\left(2\left(-k\right)ka^{2}\right)^{2}-4\left(a^{2}k^{2}+b^{2}\right)\left(-\left(b-k\right)\left(b+k\right)a^{2}\right)}}{2\left(a^{2}k^{2}+b^{2}\right)}$$

Square a^{2}\times 2k\left(-k\right).

$$x=\frac{-2\left(-k\right)ka^{2}±\sqrt{4a^{4}k^{4}-4\left(a^{2}k^{2}+b^{2}\right)\left(-\left(b-k\right)\left(b+k\right)a^{2}\right)}}{2\left(a^{2}k^{2}+b^{2}\right)}$$

Multiply -4 times b^{2}+a^{2}k^{2}.

$$x=\frac{-2\left(-k\right)ka^{2}±\sqrt{4a^{4}k^{4}+\left(-4a^{2}k^{2}-4b^{2}\right)\left(-\left(b-k\right)\left(b+k\right)a^{2}\right)}}{2\left(a^{2}k^{2}+b^{2}\right)}$$

Multiply -4b^{2}-4a^{2}k^{2} times -\left(k+b\right)\left(-k+b\right)a^{2}.

$$x=\frac{-2\left(-k\right)ka^{2}±\sqrt{4a^{4}k^{4}+4\left(b-k\right)\left(b+k\right)a^{2}\left(a^{2}k^{2}+b^{2}\right)}}{2\left(a^{2}k^{2}+b^{2}\right)}$$

Add 4a^{4}k^{4} to 4\left(b^{2}+k^{2}a^{2}\right)\left(k+b\right)\left(-k+b\right)a^{2}.

$$x=\frac{-2\left(-k\right)ka^{2}±\sqrt{4a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}}{2\left(a^{2}k^{2}+b^{2}\right)}$$

Take the square root of 4\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}.

$$x=\frac{-2\left(-k\right)ka^{2}±2\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}}{2\left(a^{2}k^{2}+b^{2}\right)}$$

Multiply 2 times b^{2}+a^{2}k^{2}.

$$x=\frac{2a^{2}k^{2}±2\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}}{2a^{2}k^{2}+2b^{2}}$$

Now solve the equation x=\frac{2a^{2}k^{2}±2\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}}{2a^{2}k^{2}+2b^{2}} when ± is plus. Add 2a^{2}k^{2} to 2\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}.

$$x=\frac{2\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+2a^{2}k^{2}}{2a^{2}k^{2}+2b^{2}}$$

Divide 2k^{2}a^{2}+2\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}} by 2b^{2}+2k^{2}a^{2}.

$$x=\frac{\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}$$

Now solve the equation x=\frac{2a^{2}k^{2}±2\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}}{2a^{2}k^{2}+2b^{2}} when ± is minus. Subtract 2\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}} from 2a^{2}k^{2}.

$$x=\frac{-2\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+2a^{2}k^{2}}{2a^{2}k^{2}+2b^{2}}$$

Divide 2k^{2}a^{2}-2\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}} by 2b^{2}+2k^{2}a^{2}.

$$x=\frac{-\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}$$

There are two solutions for x: \frac{k^{2}a^{2}+\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}}{b^{2}+k^{2}a^{2}} and \frac{k^{2}a^{2}-\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}}{b^{2}+k^{2}a^{2}}. Substitute \frac{k^{2}a^{2}+\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}}{b^{2}+k^{2}a^{2}} for x in the equation y=kx-k to find the corresponding solution for y that satisfies both equations.

$$y=k\times \frac{\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}-k$$

Multiply k times \frac{k^{2}a^{2}+\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}}{b^{2}+k^{2}a^{2}}.

$$y=\frac{\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}k-k$$

Now substitute \frac{k^{2}a^{2}-\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}}{b^{2}+k^{2}a^{2}} for x in the equation y=kx-k and solve to find the corresponding solution for y that satisfies both equations.

$$y=k\times \frac{-\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}-k$$

Multiply k times \frac{k^{2}a^{2}-\sqrt{\left(a^{2}k^{2}+b^{2}-k^{2}\right)a^{2}b^{2}}}{b^{2}+k^{2}a^{2}}.

$$y=\frac{-\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}k-k$$

The system is now solved.

$$y=\frac{\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}k-k,x=\frac{\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}\text{ or }y=\frac{-\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}k-k,x=\frac{-\sqrt{a^{2}b^{2}\left(a^{2}k^{2}+b^{2}-k^{2}\right)}+a^{2}k^{2}}{a^{2}k^{2}+b^{2}}$$

Solve for x, y (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{\left(ab\right)^{2}\left(b^{2}-k^{2}+\left(ak\right)^{2}\right)}+\left(ak\right)^{2}}{b^{2}+\left(ak\right)^{2}}\text{, }y=-\frac{k\left(\sqrt{\left(ab\right)^{2}\left(b^{2}-k^{2}+\left(ak\right)^{2}\right)}+b^{2}\right)}{b^{2}+\left(ak\right)^{2}}\text{; }x=\frac{\sqrt{\left(ab\right)^{2}\left(b^{2}-k^{2}+\left(ak\right)^{2}\right)}+\left(ak\right)^{2}}{b^{2}+\left(ak\right)^{2}}\text{, }y=\frac{k\left(\sqrt{\left(ab\right)^{2}\left(b^{2}-k^{2}+\left(ak\right)^{2}\right)}-b^{2}\right)}{b^{2}+\left(ak\right)^{2}}\text{, }&b\neq 0\text{ and }\left(a\neq -\frac{ib}{k}\text{ or }k=0\right)\text{ and }\left(a\neq \frac{ib}{k}\text{ or }k=0\right)\text{ and }a\neq 0\\x=\frac{k^{2}-b^{2}}{2k^{2}}\text{, }y=\frac{-b^{2}-k^{2}}{2k}\text{, }&\left(a=\frac{ib}{k}\text{ or }a=-\frac{ib}{k}\right)\text{ and }k\neq 0\text{ and }b\neq 0\end{matrix}\right.$$