If the lines `L_1a n dL_2` are tangents to `4x^2-4x-24 y+49=0` and are normals for `x^2+y^2=72 ,` then find the slopes of `L_1` and `L_2dot`
If the lines `L_1a n dL_2` are tangents to `4x^2-4x-24 y+49=0` and are normals for `x^2+y^2=72 ,` then find the slopes of `L_1` and `L_2dot`
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As `L_(1)andL_(2)` are normals to `x^(2)+y^(2)=72`, they must be of the form y=mx.
Sloving it with parabola `4x^(2)-4x-24y+49=0`, we have
`4x^(2)-4x-24mx+49=0`
`or4x^(2)-4x(1+6m)+49=0`
Since the lines touch the parabola,
D=0
`or16(6m+1)^(2)-16xx49=0`
`or6m+1pm7`
`i.e.,m=1or-(4)/(3)`
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