The equation of the straight line passing through the point (4$$, $$5) and making equal angles with the two straight lines given by the equations $$3x$$ – $$4y$$ – $$7$$ $$=$$ $$0$$ and $$12x$$ – $$5y$$ $$+$$ $$6$$ $$=$$ $$0$$, is

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Infinity Learn · Accepted Answer

The equation of the bisectors of the angles between the given lines $$are3x-4y-732+(-4)2=$$±$$12x-5x+6122+-52$$⇒ (39x$$ – $$52y$$ – $$91) $$=$$ ± (60x$$ – $$25y$$ $$+$$ $$30).Taking the negative sign,we get $$99x$$ – $$77y$$ – $$61$$ $$=$$ $$0$$                               (1)Taking$$ the positive sign,we get $$21x$$ $$+$$ $$27y$$ $$+$$ $$121$$ $$=$$ $$0$$                             $$(2)The$$ required lines are the lines through $$(4$$, $$5) and parallel to the bisectors of the angles between the given lines.The equation of a line parallel to (1), $$is99x$$ – $$77y$$ $$+$$ $$k1$$ $$=$$ $$0$$                                             (3)This$$ passes through $$(4$$, $$5),∴ $$99$$ (4) – $$77$$ (5) $$+$$ $$k1$$ $$=$$ $$0$$ ⇒ $$k1$$ $$=$$ –$$11$$.Putting $$k1$$ $$=$$ –$$11$$ in (3), we $$get99x$$ – $$77y$$ – $$11$$ $$=$$ $$0$$ or $$9x$$ – $$7y$$ – $$1$$ $$=$$ $$0$$.This is one of the required lines.The equation of a line parallel to (2), $$is21x$$ $$+$$ $$27y$$ $$+$$ $$k2$$ $$=$$ $$0$$                                             (4)This$$ passes through $$(4$$, $$5),∴ $$84$$ $$+$$ $$135$$ $$+$$ $$k2$$ $$=$$ $$0$$ ⇒ $$k2$$ $$=$$ –$$219$$.Putting $$k2$$ $$=$$ –$$219$$ in (4), we $$get21x$$ $$+$$ $$27y$$ – $$219$$ $$=$$ $$0$$ or $$7x$$ $$+$$ $$9y$$ – $$73$$ $$=$$ $$0$$,which is the other required line.

The equation of the straight line passing through the point (4, 5) and making equal angles with the two straight lines given by the equations 3x – 4y – 7 = 0 and 12x – 5y + 6 = 0, is

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