If a, b, c are distinct positive real numbers such that the parabolas

1.	If a, b, c are distinct positive real numbers such that the parabolas y2 = 4ax and y2 = 4c (x– b) will have a common normal, then(a) 0 < b/a - c < 1(b) b/a - c < 0(c) 1 <b/a - c < 2(d) b/a - c > 2
Answer» Correct option (d) b/a - c > 2 Explanation: Equation of normals are y = mx –2am – am³ ......(1) y = m(x – b) – 2cm – cm³ ......(2) Equation 1 and 2 are identical then –2am – am³ = –bm –2cm –cm³ / m 2a + am² = b + 2c + cm² (a– c)m² = b + 2(c – a) m² = b/a - c - 2 m = ± √b/a - c - 2 For m be real b/a - c > 2

If a, b, c are distinct positive real numbers such that the parabolas y2 = 4ax and y2 = 4c (x– b) will have a common normal, then(a) 0 < b/a - c < 1(b) b/a - c < 0(c) 1 <b/a - c < 2(d) b/a - c > 2

Answer»

Correct option (d) b/a - c > 2

Explanation:

Equation of normals are

y = mx –2am – am³ ......(1)

y = m(x – b) – 2cm – cm³ ......(2)

Equation 1 and 2 are identical then

–2am – am³ = –bm –2cm –cm³ / m

2a + am² = b + 2c + cm²

(a– c)m² = b + 2(c – a)

m² = b/a - c - 2

m = ± √b/a - c - 2

For m be real b/a - c > 2

Discussion

No Comment Found

Related InterviewSolutions

Statement1: An equation of a common tangent to the parabola `y^2=16sqrt(3)x`and the ellipse `2x^2+""y^2=""4""i s""y""=""2x""+""2sqrt(3)`.Statement 2:If the line `y""=""m x""+(4sqrt(3))/m ,(m!=0)`is a common tangent to theparabola `y^2=""16sqrt(3)x`and the ellipse `2x^2+""y^2=""4`, then m satisfies `m^4+""2m^2=""24`.(1)Statement 1 isfalse, statement 2 is true(2)Statement 1 istrue, statement 2 is true; statement 2 is a correct explanation for statement1(3)Statement 1 istrue, statement 2 is true; statement 2 is not a correct explanation forstatement 1(4)Statement 1 istrue, statement 2 is false
A normal drawn to the parabola `=4a x`meets the curve again at `Q`such that the angle subtended by `P Q`at the vertex is `90^0dot`Then the coordinates of `P`can be`(8a ,4sqrt(2)a)`(b) `(8a ,4a)``(2a ,-2sqrt(2)a)`(d) `(2a ,2sqrt(2)a)`A. `(8a,4sqrt(2)a)`B. (8a,4a)C. `(2a,-2sqrt(2)a)`D. `(2a,2sqrt(2)a)`
Find the equation of the common tangent to the curves `y^2=8x` and xy=-1.A. 3y = 9x + 2B. y = 2x + 1C. 2y = x + 8D. y = x + 2
Maximum number of common normals of `y^2=4ax and x^2=4by` is ____A. 3B. 4C. 6D. 5
Find the equation of the common tangent to the curves `y^2=8x` and xy=-1.A. 3y=9y+2B. y=2x+1C. 2y=x+8D. y=x+2
If from a point A, two tangents are drawn to parabola `y^2 = 4ax` are normal to parabola `x^2 = 4by`, thenA. `a^(2)geb^(2)`B. `a^(2)ge4b^(2)`C. `a^(2)ge8b^(2)`D. `8a^(2)geb^(2)`
The common tangent of the parabolas `y^(2)=4x" and "x^(2)=-8y,` isA. y=x+2B. y=x-2C. y=2x+3D. none of these
If a, b, c are distinct positive real numbers such that the parabolas y2 = 4ax and y2 = 4c (x– b) will have a common normal, then(a) 0 < b/a - c < 1(b) b/a - c < 0(c) 1 <b/a - c < 2(d) b/a - c > 2
Two equal parabolas have the same vertex and their axes are at right angles. The length of the common tangent to them, isA. 3aB. `3sqrt2a`C. 6aD. 2a
From an external point `P ,`a pair of tangents is drawn to the parabola `y^2=4xdot`If `theta_1a n dtheta_2`are the inclinations of these tangents with the x-axis such that `theta_1+theta_2=pi/4`, then find the locus of `Pdot`A. `x - y +1 = 0`B. `x +y - 1 = 0`C. `x - y - 1 = 0`D. `x +y + 1 = 0`

If a, b, c are distinct positive real numbers such that the parabolas y2 = 4ax and y2 = 4c (x– b) will have a common normal, then(a) 0 &lt; b/a - c &lt; 1(b) b/a - c &lt; 0(c) 1 &lt;b/a - c &lt; 2(d) b/a - c &gt; 2

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment

If a, b, c are distinct positive real numbers such that the parabolas y2 = 4ax and y2 = 4c (x– b) will have a common normal, then(a) 0 < b/a - c < 1(b) b/a - c < 0(c) 1 <b/a - c < 2(d) b/a - c > 2