Advertisements
Advertisements
प्रश्न
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
विकल्प
1
0
– 6
6
Advertisements
उत्तर
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is 6.
Explanation:
Equation of the given curves are ay + x2 = 7 .....(i)
And x3 = y .....(ii)
Differentiating eq. (i) w.r.t. x, we have
`"a" "dy"/"dx" + 2x` = 0
⇒ `"dy"/"dx" = - (2x)/"a"`
∴ m1 = `- (2x)/"a"` ......`("m"_1 = "dy"/"dx")`
Now differentiating eq. (ii) w.r.t. x, we get
3x2 = `"dy"/"dx"`
⇒ m2 = `3x^2` .....`("m"_2 = "dy"/"dx")`
The two curves are said to be orthogonal if the angle between the tangents at the point of intersection is 90°.
∴ m1 × m2 = – 1
⇒ `(-2x)/"a" xx 3x^2` = – 1
⇒ `(-6x^3)/"a"` = – 1
⇒ 6x3 = a
(1, 1) is the point of intersection of two curves.
∴ 6(1)3 = a
So a = 6
APPEARS IN
संबंधित प्रश्न
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Find the equation of the tangent to the curve x = sin 3t, y = cos 2t at
\[t = \frac{\pi}{4}\] ?
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x2 ?
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
Find the slope of the normal at the point 't' on the curve \[x = \frac{1}{t}, y = t\] ?
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.
Find the co-ordinates of the point on the curve `sqrt(x) + sqrt(y)` = 4 at which tangent is equally inclined to the axes
The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The tangent to the parabola x2 = 2y at the point (1, `1/2`) makes with the x-axis an angle of ____________.
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
Let `y = f(x)` be the equation of the curve, then equation of normal is
The slope of the tangentto the curve `x= t^2 + 3t - 8, y = 2t^2 - 2t - 5` at the point `(2, -1)` is
If the curves y2 = 6x, 9x2 + by2 = 16, cut each other at right angles then the value of b is ______.
