Advertisements
Advertisements
प्रश्न
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.
उत्तर
The equation of the given curve is y = x2 − 2x + 7
On differentiating with respect to x, we get:
`"dy"/"dx" = 2x - 2`
The equation of the line is 2x − y + 9 = 0.
2x − y + 9 = 0 ⇒ y = 2x + 9
This is of the form y = mx + c.
∴ Slope of the line = 2
If a tangent is parallel to the line 2x − y + 9 = 0, then the slope of the tangent is equal to the slope of the line.
Therefore, we have:
2 = 2x − 2
⇒ 2x = 4
⇒ x = 2
Now, x = 2
⇒ y = 4 - 4 + 7 = 7
Thus, the equation of the tangent passing through passing through (2, 7) is given by,
⇒ y - 7 = 2 (x - 2)
⇒ y - 2x - 3 = 0
Hence, the equation of the tangent line to the given curve (which is parallel to line 2x - y + 9 = 0) is y - 2x - 3 = 0.
APPEARS IN
संबंधित प्रश्न
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.
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 equation of the tangent to the curve `y = sqrt(3x-2)` which is parallel to the line 4x − 2y + 5 = 0.
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
(A) 1
(B) 2
(C) 3
(D) 1/2
If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?
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 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 points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?
Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at \[\left( ct, \frac{c}{t} \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
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 = at2, y = 2at at t = 1 ?
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
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 a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
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 ?
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that a2cos2 \[\alpha\] \[-\] b2sin2 \[\alpha\] = p2 ?
Write the equation of the normal to the curve y = x + sin x cos x at \[x = \frac{\pi}{2}\] ?
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
Find the angle of intersection of the curves y2 = x and x2 = y.
Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.
Find an angle θ, 0 < θ < `pi/2`, which increases twice as fast as its sine.
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
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
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 (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 ______.
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, From this horizontal plane is ______.
