Q.1. A line passing through (21,30) and normal to the curve y = 2√x .If m is slope of the normal then m+6 =
Ans. 1
Equation of the normal is 3 y = mx  2mm^{3}If it pass through (21,30) we have
30 = 21m 2mm^{3} ⇒ m^{3} 19m+ 30 = 0
Then m = 5, 2,3
But if m= 2 or 3 then the point where the normal meets the curve will be ( am^{2}, 2am) where the curve does not exist.
Therefore m= 5
∴ m+6 =1
Q.2. The locus of mid points of a system parallel chords with slope = 4 to the parabola 7y^{2}=25x. is 56y=k^{2} then k =
Ans. 5
It is locus of Midpoint of system of parallel chords.
Q.3. If L be the shortest distance between parabola y^{2} = 100x and circle (x – 26)^{2} + y^{2} = 1, then denotes the G.I.F. is
Ans. 5
Shortest distance between two nonintersecting curves occurs along a common normal.
Q.4. If any tangent to parabola y2 = 4x cut the coordinate axes at A and B respectively, The locus of mid point of AB is then λ is
Ans. 2
Let any tangent to parabola y^{2} = 4x is yt = x + t^{2}
Let R is (h, k)Locus is
Q.5. The equation of the line touching both the parabolas y^{2} = 4x and x^{2} = 32y is x  2y + λ = 0, then λ is
Ans. 4
Equation of tangent of parabola y^{2} = 4x isEquation (i) is also a tangent of x^{2} = 32y
thenCondition of tangency (D = 0)
From Equation (i),
Q.6. Tangents are drawn from the points on the parabola y^{2} = 8(x+4) to the parabola y2 = 4x, if local of mid point of chord of contact is again a parabola, with length of latus rectum λ, then 5λ is ....... .
Ans. 8
Let (x_{1}, y_{1}) be a point on y^{2} = 8(x + 4)
Equation of chord of contact is
2x  y_{1} y + 2x_{1} = 0, if p(h, k) be its midpoint, then its equation will be
2x  ky + k^{2}  2h = 0
Compare both k = y_{1} , 2x_{1} = k^{2}  2h
So,
Q.7. Minimum distance between the parabolas y^{2 }– 4x – 8y + 40 = 0 and x^{2 }– 8x – 4y + 40 = 0 is√λ. Then the value of λ is
Ans. 2
Since two parabolas are symmetrical about y = x
Minimum distance is distance between tangents to the parabola and parallel to y = x.
Differentiating x^{2}  8x  4y + 40 = 0 w.r.t x, we get 2x  8  4y' = 0Corresponding point on (y  4)2 = 4(x  6) is (7, 6) so minimum distance = 2 .
Q.8. The shortest distance between the parabolas y^{2} = x – 1 and
then the value of numerical quantity k must be.
Ans. 4
Since both curves are symmetrical about the line y = x distance between any pair of
points = 2(distance of (t, t^{2} + 1) on the parabola y = x^{2} + 1 from y – x = 0)
The minimum value of ⇒ Minimum distance
Q. 9. If a focal chord of y^{2} =16x is a tangent to the circle (x6)^{2} + y^{2 }= 2, then the positive value of the slope of this chord is
Ans. 1
Equation of parabola y^{2} =16x, focus (4, 0).
The equation of tangents of slope m to the circle
If this tangents pass through the focus i.e. (4,0), then
2m^{2} = 2⇒m = ±1∴ the positive value of m = +1.
Q.10. Circle (x  1)^{2} + (y  1)^{2} = 1 touches X and Y axis at A and B respectively. A line y = mx intersects this circle at P and Q. If area of ΔPQB is maximum then find the value of .(where [.] denotes the greatest integer function).
Ans. 1
Area of which is maximum where m = 1/√3
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