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Circle question

2024 · 8 Apr · Shift 2 · Q49
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  5. /2024 · 8 Apr · Shift 2 · Q49

Circle question

2024 · 8 Apr · Shift 2 · Q49

JEE MainMathematicsCircleMCQ+4 / −1
If the image of the point (−4,5)(-4,5)(−4,5) in the line x+2y=2x+2 y=2x+2y=2 lies on the circle (x+4)2+(y−3)2=r2(x+4)^2+(y-3)^2=r^2(x+4)2+(y−3)2=r2, then rrr is equal to:
  1. A
    2
  2. B
    3
  3. C
    4
  4. D
    1
View written solutionFree

Correct answer: A

  1. Given data
  • Point: P(−4,5)P(-4,5)P(−4,5)
  • Mirror line: x+2y=2x+2y=2x+2y=2, i.e. x+2y−2=0x+2y-2=0x+2y−2=0
  • Circle: (x+4)2+(y−3)2=r2(x+4)^2+(y-3)^2=r^2(x+4)2+(y−3)2=r2

We need the image of PPP in the line x+2y=2x+2y=2x+2y=2, and that image lies on the circle.

  1. Reflection formula

For a point (x1,y1)(x_1,y_1)(x1​,y1​) reflected in the line ax+by+c=0,ax+by+c=0,ax+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)​).

Here, a=1,b=2,c=−2,(x1,y1)=(−4,5).a=1,\quad b=2,\quad c=-2,\quad (x_1,y_1)=(-4,5).a=1,b=2,c=−2,(x1​,y1​)=(−4,5).

First compute: ax1+by1+c=1(−4)+2(5)−2=−4+10−2=4.ax_1+by_1+c=1(-4)+2(5)-2=-4+10-2=4.ax1​+by1​+c=1(−4)+2(5)−2=−4+10−2=4.

Also, a2+b2=12+22=5.a^2+b^2=1^2+2^2=5.a2+b2=12+22=5.

So the reflected point is

=\left(-4-\frac{8}{5},\;5-\frac{16}{5}\right) =\left(-\frac{28}{5},\;\frac{9}{5}\right).$$ Let this image be $P'\left(-\frac{28}{5},\frac{9}{5}\right)$. --- 3. **Use the circle equation** The circle is $$(x+4)^2+(y-3)^2=r^2,$$ so its center is $$(-4,3).$$ Since $P'$ lies on the circle, $$r^2=\left(-\frac{28}{5}+4\right)^2+\left(\frac{9}{5}-3\right)^2.$$ Now simplify: $$-\frac{28}{5}+4=-\frac{28}{5}+\frac{20}{5}=-\frac{8}{5},$$ $$\frac{9}{5}-3=\frac{9}{5}-\frac{15}{5}=-\frac{6}{5}.$$ Thus, $$r^2=\left(-\frac{8}{5}\right)^2+\left(-\frac{6}{5}\right)^2 =\frac{64}{25}+\frac{36}{25} =\frac{100}{25}=4.$$ Therefore, $$r=2.$$ --- 4. **Check options** - A: $2$ ✅ - B: $3$ ❌ - C: $4$ ❌ - D: $1$ ❌ So the correct option is **A**.
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