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Standard Equation of Ellipse

The lines L...

Question

The lines $L_{1}$ and $L_{2}$ denoted by $3 x^{2} + 10 x y + 8 y^{2} + 14 x + 22 y + 15 = 0$ intersect at the point $P$ and have gradients $m_{1}$ and $m_{2}$ respectively. The acute angles between them is $θ$ . Which of the following relations hold good ?

m_{1} + m_{2} = 5 / 4

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m_{1} m_{2} = 3 / 8

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Acute angle between

L_{1}

and

L_{2}

s i n^{- 1} (\frac{2}{5 \sqrt{5}}) .

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Sum of the abscissa and ordinate of the point

P

- 1.

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Solution

The correct options are
B $m_{1} m_{2} = 3 / 8$
C Acute angle between $L_{1}$ and $L_{2}$ is $s i n^{- 1} (\frac{2}{5 \sqrt{5}}) .$
D Sum of the abscissa and ordinate of the point $P$ is $- 1.$
$3 x^{2} + 10 x y + 8 y^{2} + 14 x + 22 y + 15 = 0$ ----- $(1)$

$3 x^{2} + 10 x y + 8 y^{2} = 0$

$3 x^{2} + 6 x y + 4 x y + 8 y^{2} = 0$

$3 x (x + 2 y) + 4 y (x + 2 y) = 0$

$(3 x + 4 y) (x + 2 y) = 0$

$(4 y + 3 x + c_{1}) (2 y + x + c_{2}) = 0$

$4 y c_{2} + 3 x c_{2} + 2 y c_{1} + x c_{1} + 8 y^{2} + 3 x^{2} + 10 x y + c_{1} c_{2} = 0$ ----- $(2)$

$3 x^{2} + 10 x y + 8 y^{2} + 4 y c_{2} + 3 x c_{2} + 8 y c_{1} + x c_{1} + c_{1} c_{1} = 0$

on comparing equation $(1)$ and $(2)$

$3 c_{2} + c_{1} = 14$ ---- $(3)$

$4 c_{2} + 2 c_{1} = 22$ ---- $(4)$

From equation $3$ and $4$

$c_{1} = 5, c_{2} = 3$

$3 x^{2} + 10 x y + 8 y^{2} = 0$

$8 y^{2} + 10 x y + 3 x^{2} = 0$

$y^{2} + \frac{10}{8} x y + \frac{3}{8} x^{2} = 0$

$y^{2} + (m_{1} + m_{2}) x y + m_{1} m_{2} x^{2} = 0$

$m_{1} + m_{2} = - \frac{b}{a} = - \frac{10}{8} = - \frac{5}{4}$ ----- $(5)$

$m_{1} m_{2} = \frac{c}{a} = \frac{3}{8}$ ---- $(6)$

$tan θ = | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} | = | \frac{- \frac{1}{2} + \frac{3}{4}}{1 + \frac{3}{8}} | = \frac{2}{11}$

$sin θ = \frac{2}{5 \sqrt{5}}$

$4 y + 3 x + 5 = 0$ ----- $(7)$

$2 y + x + 3 = 0$ ----- $(8)$

then,

$x = 1, y = - 2$

Sum $= x + y = - 1$

Hence the correct options are $B, C$ and $D$ .

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The lines L1 and L2 denoted by 3x2+10xy+8y2+14x+22y+15=0 intersect at the point P and have gradients m1 and m2 respectively. The acute angles between them is θ. Which of the following relations hold good ?

The lines $L_{1}$ and $L_{2}$ denoted by $3 x^{2} + 10 x y + 8 y^{2} + 14 x + 22 y + 15 = 0$ intersect at the point $P$ and have gradients $m_{1}$ and $m_{2}$ respectively. The acute angles between them is $θ$ . Which of the following relations hold good ?