Question 65

The combined equation of the two lines $$ax + by + c = 0$$ and $$a'x + b'y + c' = 0$$ can be written as $$(ax + by + c)(a'x + b'y + c') = 0$$. The equation of the angle bisectors of the lines represented by the equation $$2x^2 + xy - 3y^2 = 0$$ is

Given: $$2x^2 + xy - 3y^2 = 0$$

Comparing with the standard form, $$ax^2 + 2hxy + by^2 = 0$$:

$$a = 2, \quad b = -3, \quad 2h = 1 \implies h = \frac{1}{2}$$

Substituting the values into the angle bisector formula:

$$\frac{x^2 - y^2}{2 - (-3)} = \frac{xy}{\frac{1}{2}}$$

$$\frac{x^2 - y^2}{5} = 2xy$$

$$x^2 - y^2 = 10xy \implies x^2 - y^2 - 10xy = 0$$

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