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Question

The angle between the lines joining the origin to the points of intersection of the line y = 3x + 2with the curve x2 + 2xy + 3y2 + 4x + 8y = 11, is

A
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B
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C
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D
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Solution

The correct option is C
The equation of the given straight line is
Y=3x+2y3x2......(i)
The equation of the given curve is
x2+2xy+3y2+4x+8y11=0........(ii)
The combined equation of the straight lines joining the origin to the points of intersection of Eqs. (i) and (ii) is a homogeneous equation of second degree obtained with the help of Eqs. (i) and (ii). Making the Eq. (ii) homogeneous of the second degree in x and y with the help of Eq. (i), we get
x2+2xy+3y2+4x(y3x2)+8y(y3x2)11(y3x2)=04x2+8xy+12y2+2(8y212x220xy)11(y26xy+9x2)=07x22xyy2=0
This is the required equation. On comparing the equation with ax2+2hxy+by2=0, we obtain
a = 7, b = −1 and h = −1.
Let θ be the required angle. Then,
tanθ=2h2aba+btanθ=21+7a+bθ=tan(223)
Hence, (c) is the correct answer.

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