1.

Let from the point P(alpha, beta), tangents are drawn to the parabola y^(2)=4x, including the angle 45^(@) to each other. Then locus of P(alpha,beta) is(are)

Answer»

<P>a circle with centre (-3,0)
an ellipse with centre(-3,0)
a RECTANGULAR hyperbola with centre(3,0)
a rectangular hyperbola with centre (-3,0)

Solution :Any tangent to `y^(2)=4X` is
`y=mx+(1)/(m)rArr ` `-BETA m +1 = 0 " " ("As" , (P (alpha, beta) "lies on it " )`
Here `(m_(1)+m_(2))=(beta)/(alpha)` and `m_(1).m_(2)=(1)/(alpha)`.
Now, `tan 45^(@)=|(m_(1)+m_(2))/(1+m_(1)m_(2))|rArr (m_(1)+m_(2))^(2)-4m_(1)m_(2)=(1+m_(1)m_(2))^(2)rArr((beta)/(alpha))^(2)-(4)/(alpha)=(1+(1)/(alpha))^(2)`
`rArr beta^(2)-4 alpha=(alpha+1)^(2)rArr(alpha+3)^(2)-beta^(2)=8`
`(x+3^(2))-y^(2)=8`, which is a rectangular hyperbola with centre (-3,0).


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