Advertisements
Advertisements
प्रश्न
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Advertisements
उत्तर
The slope of the given line y = 4x − 5 is 4
\[y^2 = a x^3 + b . . . \left( 1 \right)\]
\[2y \frac{dy}{dx} = 3a x^2 \]
\[ \Rightarrow \frac{dy}{dx} = \frac{3a x^2}{2y}\]
\[\text { Slope of tangent }= \left( \frac{dy}{dx} \right)_\left( 2, 3 \right) =\frac{12a}{6}=2a\]
\[\text { Given that }\]
\[\text { Slope of tangent = slope of given line }\]
\[2a = 4\]
\[ \Rightarrow a = 2\]
\[\text { Substituting this and }x= 2,y= 3 \text{ in (1), we get }\]
\[9 = 16 + b\]
\[ \Rightarrow b = - 7\]
\[\text { Hence, a}= 2 \text { and }b = - 7\]
APPEARS IN
संबंधित प्रश्न
Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.
The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.
Find the equations of the tangent and normal to the hyperbola `x^2/a^2 - y^2/b^2` at the point `(x_0, y_0)`
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 ?
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4x at (1, 2) ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?
Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4x − 2y + 5 = 0 ?
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
Write the equation of the normal to the curve y = cos x at (0, 1) ?
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
The curves y = aex and y = be−x cut orthogonally, if ___________ .
Find the angle of intersection of the curves y2 = x and x2 = y.
The tangent to the curve given by x = et . cost, y = et . sint at t = `pi/4` makes with x-axis an angle ______.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
Show that the line `x/"a" + y/"b"` = 1, touches the curve y = b · e– x/a at the point where the curve intersects the axis of y
If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.
The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.
For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
Find a point on the curve y = (x – 2)2. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
Find the points on the curve `y = x^3` at which the slope of the tangent is equal to the y-coordinate of the point
If (a, b), (c, d) are points on the curve 9y2 = x3 where the normal makes equal intercepts on the axes, then the value of a + b + c + d is ______.
The number of values of c such that the straight line 3x + 4y = c touches the curve `x^4/2` = x + y is ______.
