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Straight Lines and Pair of Straight Lines question

2020 · 6 Sep · Shift 2 · Q19
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Straight Lines and Pair of Straight Lines question

2020 · 6 Sep · Shift 2 · Q19

JEE MainMathematicsStraight Lines and Pair of Straight LinesMCQ+4 / −1
Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :
  1. A
    (115,285)\left( {{{11} \over 5},{{28} \over 5}} \right)(511​,528​)
  2. B
    (295,115)\left( {{{29} \over 5},{{11} \over 5}} \right)(529​,511​)
  3. C
    (295,85)\left( {{{29} \over 5},{8 \over 5}} \right)(529​,58​)
  4. D
    (85,295)\left( {{8 \over 5},{{29} \over 5}} \right)(58​,529​)
View written solutionFree

Correct answer: A

  1. Equation of the line

The line has x-intercept 333 and y-intercept 111, so its intercept form is

x3+y1=1\frac{x}{3}+\frac{y}{1}=13x​+1y​=1

which simplifies to

x+3y−3=0.x+3y-3=0.x+3y−3=0.

  1. Reflection of a point across a line

We need the image of P(−1,−4)P(-1,-4)P(−1,−4) in the line

ax+by+c=0,ax+by+c=0,ax+by+c=0,

where here

a=1,b=3,c=−3.a=1,\quad b=3,\quad c=-3.a=1,b=3,c=−3.

For reflection of (x1,y1)(x_1,y_1)(x1​,y1​) across ax+by+c=0ax+by+c=0ax+by+c=0, the image is

(x1−2a(ax1+by1+c)a2+b2,  y1−2b(ax1+by1+c)a2+b2).\left(x_1-\frac{2a(ax_1+by_1+c)}{a^2+b^2},\; y_1-\frac{2b(ax_1+by_1+c)}{a^2+b^2}\right).(x1​−a2+b22a(ax1​+by1​+c)​,y1​−a2+b22b(ax1​+by1​+c)​).

  1. Substitute the point

For P(−1,−4)P(-1,-4)P(−1,−4),

ax1+by1+c=1(−1)+3(−4)−3=−1−12−3=−16.ax_1+by_1+c = 1(-1)+3(-4)-3=-1-12-3=-16.ax1​+by1​+c=1(−1)+3(−4)−3=−1−12−3=−16.

Also,

a2+b2=12+32=10.a^2+b^2=1^2+3^2=10.a2+b2=12+32=10.

Thus the reflected point is

x′=−1−2(1)(−16)10=−1+3210=−1+165=115,x'=-1-\frac{2(1)(-16)}{10}=-1+\frac{32}{10}=-1+\frac{16}{5}=\frac{11}{5},x′=−1−102(1)(−16)​=−1+1032​=−1+516​=511​,

y′=−4−2(3)(−16)10=−4+9610=−4+485=285.y'=-4-\frac{2(3)(-16)}{10}=-4+\frac{96}{10}=-4+\frac{48}{5}=\frac{28}{5}.y′=−4−102(3)(−16)​=−4+1096​=−4+548​=528​.

So the image is

(115,285).\left(\frac{11}{5},\frac{28}{5}\right).(511​,528​).

  1. Match with options

This corresponds to Option A.

  1. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

Hence, the derived answer agrees with the stored answer.

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