If angle between the lines represented by $ax^2+2hxy+by^2=0$ is equal to the angle between the lines represented by $2x^2-5xy+3y^2=0$, then show that $100(h^2-ab)=(a+b)^2$.

If angle between the lines represented by ax²+2hxy+by²=0 is equal to the angle between the lines represented b

Question 130 of 139

ax^2+2hxy+by^2=0

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2022Subjective· 3mImportance★★★★★

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Equate $\tan\theta$ for the given pair with $\tan\theta$ of $2x^2-5xy+3y^2=0$ .

For $ax^2+2hxy+by^2=0$ : $\tan\theta = \left|\dfrac{2\sqrt{h^2-ab}}{a+b}\right|$

For $2x^2-5xy+3y^2=0$ : $a'=2,\ 2h'=-5\Rightarrow h'=-\dfrac52,\ b'=3$

$\tan\theta' = \left|\dfrac{2\sqrt{(h')^2-a'b'}}{a'+b'}\right| = \left|\dfrac{2\sqrt{\frac{25}4-6}}5\right| = \left|\dfrac{2\sqrt{\frac14}}5\right| = \dfrac{2\cdot\frac12}5 = \dfrac15$

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Q.If angle between the lines represented by $ax^2+2hxy+by^2=0$ is equal to the angle between the lines represented by $2x^2-5xy+3y^2=0$ , then show that $100(h^2-ab)=(a+b)^2$ .

Concept understanding — Angle Between a Pair of Lines Through the Origin

Q.If angle between the lines represented by ax2+2hxy+by2=0ax^2+2hxy+by^2=0ax2+2hxy+by2=0 is equal to the angle between the lines represented by 2x2−5xy+3y2=02x^2-5xy+3y^2=02x2−5xy+3y2=0, then show that 100(h2−ab)=(a+b)2100(h^2-ab)=(a+b)^2100(h2−ab)=(a+b)2.

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Q.If angle between the lines represented by $ax^2+2hxy+by^2=0$ is equal to the angle between the lines represented by $2x^2-5xy+3y^2=0$ , then show that $100(h^2-ab)=(a+b)^2$ .

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