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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 2
The general equation of a line is ax+by+c=0
Distance between the line and the point (x1,y1) is given by
∣ ∣ax1+by1+ca2+b2∣ ∣
Let us find the number of lines passing through the point (4,5) whose slope is m
y+5=m(x4)
mxy4m5=0 ..(1)
Distance between the point (2,3) and the line (1) is given by
2m34m51+m2=12
6m81+m2=12
3m41+m2=6
3m4=6(1+m2) by squaring on both sides,we get
9m2+16+24m3636m2=0
27m224m20=0 is quadratic in m where a=27,b=24,c=20
Discriminant=Δ=b24ac=(24)24×27×20=(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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