Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Given that y = b · e– x/a, the equation of curve and `x/"a" + y/"b"` = 1, the equation of line.
Let the coordinates of the point where the curve intersects the y-axis be (0, y1)
Now differentiating y = b · e– x/a both sides w.r.t. x, we get
`"dy"/"dx" = "b" * "e"^((-x)/"a") (- 1/"a")`
= `- "b"/"a" * "e"^((-x)/"a")`
So, the slope of the tangent, m1 = `- "b"/"a" * "e"^((-x)/"a")`
Differentiating `x/"a" + y/"b"` = 1 both sides w.r.t. x, we get
`1/"a" + 1/"b" * "dy"/"dx"` = 0
So, the slope of the line, m2 = ` (-"b")/"a"`.
If the line touches the curve, then m1 = m2
⇒ `(-"b")/"a" * "e"^((-x)/"a") = (-"b")/"a"`
⇒ `"e"^((-x)/"a")` = 1
⇒ `(-x)/"a" log "e"` = log 1 .....(Taking log on both sides)
⇒ `(-x)/"a"` = 0
⇒ x = 0
Putting x = 0 in equation y = `"b" * "e"^((-x)/"a")`
⇒ y = b · e0 = b
Hence, the given equation of curve intersects at (0, b) i.e. on y-axis.
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.
Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.
Find the equation of all lines having slope 2 which are tangents to the curve `y = 1/(x- 3), x != 3`
Find the equations of all lines having slope 0 which are tangent to the curve y = `1/(x^2-2x + 3)`
Find points on the curve `x^2/9 + "y"^2/16 = 1` at which the tangent is parallel to x-axis.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x3 at (1, 1)
Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]
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 equation of the normal to curve y2 = 4x at the point (1, 2).
Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.
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 points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Find the equation of the tangent and the normal to the following curve at the indicated points:
x = 3cosθ − cos3θ, y = 3sinθ − sin3θ?
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?
Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
Find the equation of tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π that are parallel to the line x + 2y = 0.
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan `"t"/2`)} at the point ‘t’ is ____________.
Find points on the curve `x^2/9 + "y"^2/16` = 1 at which the tangent is parallel to y-axis.
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining `(0, 3/2)` and `(1/2, 2)`, then ______.
An edge of variable cube is increasing at the rate of 3 cm/s. The volume of the cube increasing fast when the edge is 10 cm long is ______ cm3/s.
If m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
