Question

# Number of lines can be drawn through the point (4,−5) so that its distance from (−2,3) will be equal to 12

A
0
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B
1
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C
2
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D
Infinite
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Solution

## The correct option is C 2The general equation of a line is ax+by+c=0Distance between the line and the point (x1,y1) is given by∣∣ ∣∣ax1+by1+c√a2+b2∣∣ ∣∣ Let us find the number of lines passing through the point (4,−5) whose slope is m⇒y+5=m(x−4)⇒mx−y−4m−5=0 ..(1)Distance between the point (−2,3) and the line (1) is given by∣∣∣−2m−3−4m−5√1+m2∣∣∣=12⇒∣∣∣−6m−8√1+m2∣∣∣=12⇒∣∣∣−3m−4√1+m2∣∣∣=6⇒−3m−4=6(√1+m2) by squaring on both sides,we get⇒9m2+16+24m−36−36m2=0⇒27m2−24m−20=0 is quadratic in m where a=27,b=−24,c=−20Discriminant=Δ=b2−4ac=(−24)2−4×27×−20=(24)2+4×27×20>0⇒ Roots are real and distinct∴ there exists two roots m1 and m2 which are real and distinct∴ there exists two different lines with different slopes.Hence,number of lines =2

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